I can't help but notice word pairs between Arabic and some European languages that seem to be cognate, such as: Sabah "seven"/ German Sieben also "seven" Ktab "book"/ English tablet (both something written on) hadith "saying"/ Italian ditto "say" and to me the most obvious kafir literally "one who covers/English cover If the answer to my question is "no", then how could these word pairs exist?
If you really want to pursue this line of enquiry you need to compare proto-Afro-Asiatic with proto-Indo-European. You cannot just compare Arabic with English or Italian. Taking the etymology even a small step further demolishes most of your examples. E.g. Italian detto (not “ditto”) comes from Latin dictum, which does not look at all like Arabic ḥadīϑ. The only one of your examples that is even debatable is the word for “seven”, where Semitic *šabʽ has been compared with Indo-European *septṃ, either as evidence for a “Nostratic” super-family or else for an ancient borrowing in one direction or the other. But even in this case, coincidence is the most likely explanation.
There's this controversial hypothesis about a genetic relationship between Indo-European and Semitic languages. The Wikipedia article that deals with it concedes that it "has never been widely accepted by contemporary linguists in modern times". At the end it tries to salvage the hypothesis by moving the goalposts:
The Indo-Semitic hypothesis has thus undergone a paradigm shift. From Lepsius in 1836 through the mid-20th century, the question asked was whether Indo-European and Semitic are related or unrelated, and in attempting to answer this question Indo-European and Semitic were compared directly. This now appears naive, and the relevant units of comparison instead appear to be Eurasiatic and Afroasiatic, the immediate precursors of Indo-European (controversially) and Semitic (uncontroversially). This revised schema still has a long road to go if it is to win general acceptance from the linguistic community.
So, in short: no, there's no generally accepted evidence of a genetic relationship between Indo-European and Semitic languages.
Regarding the second question: how could these word pairs exist? The thing is, unless the pairs can be shown to be phonetically related in a systematic way, no amount of them can be used to infer a genetic relationship. In other words, they are just coincidences.
If you look for them, in fact, you'll find that there are many more such word pairs, not only between English and Arabic but between any pair of languages, of whatever family you might choose. There's a very thorough article on this ("How likely are chance resemblances between languages?") in case you find it hard to believe. The trick is relaxing the rules for considering word x cognate to word y (or starting without rules altogether).
So, I've seen many answers to this question, but few which actually make reference to specific vocabulary of Proto-Indo-European and Proto-Semitic. I'm not terribly acquainted with PIE, but I am familiar with Proto-Semitic and the Semitic languages. The ultimate answer is no, there is no direct linguistic connection between Afroasiatic and Indo-European, but there WAS very early linguistic and cultural contact between PIE and a Semitic-speaking population.
As far as the comparative method is concerned: no, there is no genetic affiliation between Indo-European and Afro-Asiatic. Only speculative "big tent" macrofamilies like Nostratic and Proto-World connect Indo-European to Afro-Asiatic--although even among Nostraticists, Afro-Asiatic is not always included. Whatever prehistoric relationship might exist between Afro-Asiatic and Indo-European, it's ancient enough to have been eroded far beyond recognition. The idea of a connection between IE and Semitic has been attractive since the earliest days of historical linguistics, mostly out of a desire of Christian European scholars to connect their languages to Hebrew, which they considered to be the "original language", and therefore proof of the biblical story of the Tower of Babel. Of course, this has not borne fruit, and we know with extreme certainty that Hebrew is far from the original language--in fact, as far as the Semitic languages go, Hebrew is among the most divergent and innovative, at least phonologically. You have reason to be skeptical of claims connecting Indo-European to Semitic; they are very often motivated more by religious dogmas than genuine academic or historical interest.
That being said, there might in fact be a number of words of common origin shared between Proto-Semitic and Proto-Indo-European! The most famous is PS *θawrum 'bull, ox', compared to PIE *táwros 'id.'. Others include:
- PS *karnum 'horn' : PIE *ḱer-(h2/w-) 'id.'
- PS *ɡ-ʕ-w 'to moo; bellow' : PIE *ɡʷṓw- 'cow'
- PS *ʔalapum 'bull, cattle' : PIE *lāp- 'cattle'
- PS *barrum 'wheat, grain' : PIE *bʰar- 'barley'
- PS *ɡadyum 'goat' : PIE *ɡʰayd- 'id.'
- PS *l-k-ʔ 'to gather, collect' : PIE *leɡ́- 'to gather'
- PS *k-r-y 'to trade' : PIE *kʷrey(h2)- 'to pay (in livestock)'
- PS *ħiθʼwum 'arrow' : PIE *h1ish2- 'id.'' [EDIT 6/16/2021: This one is dubious; the root *ħ-θʼ-w only means 'arrow, dart' in poetic Hebrew (ħēsˤ); in other Semitic languages including non-poetic Hebrew, it means something like 'to split'.]
- PS *p-l-kʼ 'to split apart; axe' : PIE *peleḱus 'axe'
- PS *sabʕatum 'seven' : PIE *septm̥ 'id.'
There's at least one PIE word, *h2eǵros 'field', which is possibly ultimately from Sumerian agar 'irrigated territory; grainfield'. Since no hypothesized Indo-European homeland would put PIE in contact with Sumerian, this can only have entered PIE by way of Semitic, which borrowed numerous Sumerian loanwords in the post-Proto-Semitic phase; compare Ge'ez hagar 'arable field' and Aramaic hăɣar 'village'.
With the very odd exception of the PIE word for 'seven' (and potentially also 'six'), shared etymologies between Proto-Semitic and Proto-Indo-European overwhelmingly have to do with the natural world, agriculture, and cattle husbandry. It seems that Proto-Indo-European was in contact with a Semitic-speaking population who introduced such terms and technologies to the Proto-Indo-Europeans.
Now, it's highly unlikely that Proto-Semitic itself is the source of Semitic words in Proto-Indo-European. Nor is it likely that Proto-Indo-European is the donor language of these terms into Semitic rather than vice versa. By the time we think Proto-Indo-European was spoken (ca. 4500 BCE), Proto-Semitic had most likely already diversified into its descendant branches for several centuries. Furthermore, if there are indeed Sumerian loanwords in Proto-Indo-European, those would inevitably have had to enter PIE by way of a post-PS Semitic language. The homeland of the Proto-Semites is not known, but it's not generally placed in the vicinity of the Pontic Steppe, which is the most accepted hypothesis for the Indo-European homeland. All of this points to the idea that there was a population of Semitic-speaking peoples--but not Proto-Semitic-speaking peoples--who migrated northwards, interacting with the Proto-Indo-Europeans and donating vocabulary relating to agriculture and animal husbandry. And, for some reason, the number 7. But, PIE apparently borrowed words for other numbers from outside sources too, so...I guess the Proto-Indo-Europeans were great at horses, not math. This hypothetical, unattested branch of Semitic is called "Balkan Semitic" in some literature.
As for Proto-Indo-European influence on Semitic, there don't seem to be any Proto-Indo-European words which are attested in all branches of Semitic. Individual words from post-PIE languages entered various languages of the Middle East, such as words relating to horse husbandry in Akkadian and Hurrian, but otherwise there's no identifiable PIE substrate in the vocabulary of Proto-Semitic. There is precisely one word found in Proto-West-Semitic (one of the branches of Proto-Semitic) which is most likely borrowed from PIE, that being *waynum 'wine; (grape) vine', probably borrowed from PIE *wéyh1ō 'id.'. This asymmetrical situation makes sense considering the fact that PS is many centuries older than PIE, and seems to bolster the hypothesis of the 'Balkan Semitic' branch which could have donated vocabulary to PIE and perhaps themselves traded wine products into the West Semitic-speaking peoples.
So that's about as much as we know regarding language contact and relations between Semitic and Proto-Indo-European. But of course, the later Indo-European languages interacted much more freely with the Semitic languages and borrowed words from them, and vice versa. The spread of Islam donated Arabic vocabulary to languages like Persian (whence into Indo-Aryan languages like Hindustani and Punjabi), Greek (directly or indirectly via Turkish or Persian), Spanish, Albanian, Armenian (directly or indirectly via Turkish or Persian), and so on. Armenian shows a number of Aramaic loanwords. Hebrew donated numerous words to the Christian world, mostly via Greek and Latin, particularly in the form of names. Jewish communities in Europe and beyond created new sociolects (and in some cases eventually languages) of the languages of the goyish community, infused with Hebrew and Aramaic vocabulary, giving rise to languages like Yiddish, Ladino, Bukhori, Juhuri, and various Judaized languages like Judeo-Greek, Judeo-Italian, Judeo-Persian, and so on. And of course in modern times, many words of Arabic origin have entered English via news and cultural diffusion. And for their part, Semitic languages have borrowed from Indo-European languages--most especially colonial European languages, especially English and Italian in the modern age, but Greek was a significant donor of vocabulary in Arabic and Medieval Hebrew. Maltese can almost be thought of as an 'inverted Yiddish': a Semitic language (Arabic) so infused with Indo-European (Italian) vocabulary and influence as to become a unique language in its own right. Even in ancient times, Indo-Aryan vocabulary for certain foreign goods was entering Semitic vocabulary via Dravidian by way of Tamil traders; the Hebrew scriptures have words like karpas 'expensive cotton fabric', which is ultimately a loanword from Sanskrit karpāsa 'cotton'.
Most of the information about Proto-Indo-European that I've used in this answer comes from "Foreign elements in the Proto-Indo-European vocabulary" by Rasmus Gudmundsen Bjørn (2017). He compiles a large number of proposed or potential cognates between PIE and Afroasiatic/Sumerian, along with the Uralic and Caucasian languages. I've taken the liberty of tweaking some of the Proto-Semitic etymologies given therein, which are quoted directly from various sources at various times in our understanding of Proto-Semitic, and using different transcription systems.
All languages are related: there's a limited inventory of phonological, lexical, and syntactic features of which any particular language has a subset.
But that's surely not what you're asking. I suppose you're asking if there is some common cultural chain from long in the past that connects the two, closer than other language groups.
A handful of pairs, some from different pairs of language, is a bit of cherry picking. You really need to show a systematic comparison.
Like with much language history questions, we weren't there at the time so we have to use comparisons of current language to check. Phonology changes so much over the years that if two languages seem to share a small handful of words very closely it is most likely a matter of a recent borrowing.
Even some very large over-arching parameters of a language, eg left vs right branching or pro-drop, can change between historically very near languages so that is not a good single measure of closeness.
But there is the vague idea based on vague commonalities like general word formation and general sentence structure (called the Nostratic hypothesis) that seem to relate the European and 'Uralic' languages together (but not Bantu), and the Sino-Tibetan and American languages together. Very vaguely. So there are some very vague commonalities between IE and Semitic that are not exactly done for the most part in other language groups. What those vagaries are... well, sometimes Afro-Asiatic (including Semitic) is included and sometimes not. In the end, the hypothesis is not considered crazy exactly, just that the evidence for it is not overwhelmingly convincing to most linguists. That is, it's pretty controversial.
All that said, there are a lot of latter day Semitic borrowings into IE languages via biblical scholarship on the one hand and the Ashkenazi Jewish diaspora in Europe on the other. In English, check out the American Heritage Dictionary appendix on Semitic Roots of English words. By 'a lot' is not even 1% of words, but the number is non-trivial.
Chance coincidences like this aren't uncommon. For example, there's an African language (whose name escapes me sadly, if anyone else can give the name it would be greatly appreciated) whose word for 'dog' is 'dog'. This is pure coincidence, the two words in fact have separate ancestries that just so happened to converge on each other over time.
There's thousands of languages on this planet, but only a few hundred phonemes, and taking into account how humans tend to see some (though not all) foreign sounds as sounding like native phonemes, coincidences like this happen far more often than you think. For instance, in Japanese the word for 'name' is 'namae'. They aren't related, in fact, 'namae' is a compound word made from native roots. Also, the similarity is more in how its spelled in the Latin alphabet than its pronunciation.
Like others have stated, there is no evidence that the Semitic and Indo-european families are related. Some think there may have been loan words between the two proto-languages, but that's the closest to a relationship that anyone seriously thinks exists between the two language families.
Also, there's a number of suspected relationships that aren't (currently) accepted by the linguistic community. Some seriously argue that Japanese and Korean are related (both are accepted to be language isolates). This mostly comes from their striking similarities in grammar and the huge number of Chinese loan words they took in the past. There are a handful of native words in the two that look like they could be related, but again its generally accepted to be just a coincidence.
To be considered related, you have to find far more words that look similar. If you want an example, go look at the evidence that shows that Hindi/urdu is part of the Indo-european family. You can find thousands of Hindi/urdu words that are obviously similar to English words with related meanings. Even their pronouns are obviously related! These are the kinds of similarities that linguists accept as indicating a relationship between two languages, not the few pieces of evidence that you suggest. Besides, if they were related you would expect to see far more words with similarities, rather than vocabularies where 99% of the words bear no resemblance what-so-ever.
Of course, you could possibly consider them related if you assume that all current languages have a single common ancestor in the ancient past. But in reality, there's no way to confirm or deny this. We don't know if there was ever a single proto-language that all languages descend from. Language may have appeared in multiple different populations independently of each other. Really, the origins of language are impossible to determine, and its commonly accepted that its just something we'll never know due to the lack of written evidence.
What is presented here is the standard view, but it might be partly otherwise. Nostratic might actually have been a dialectical continuum in a large area in the Middle East, without any language boundaries, but still with layers having distinct features.
Uralic and Altaic branches might have originated from early migrations form the north of that continuum.
The southern Afro-asiatic languages possibly originated early from the southern parts of this continuum.
Indo-european and Semitic possibly developed developed by migrations from two adjacent mid levels in that continnuum at a later time.
This picture explains that Indo-european in many respects is very Semitic-like and in other very Uralic-like. In this interpretation Afro-asiatic was nothing but the southern half of that continuum, and Eurasitic nothing but the Northern half.
At some very early period Nostratic was probably a more compact and homogenic language, but after developing into a dialectical continuum, several layers from south to north developed with many of the traits we can see today in the different sub-brtanches like Indo-european and Semitic.
Indo-Eurpean languages are also realted to Semitic languages, here a few examples:
EARTH < ארץ AReTS, earth
PALSY < פלץ PaLaTS, to tremble or shake
PANE < פני PiNaY, “face" or the "surface of" the waters
(EN)CASE < כסה Ka$aH, to cover, conceal, encase
ENDOW < נדב NaDa(V), to donate