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).
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.
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.
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