I'm an speaker of a dialect of the Azerbaijani language, Tabriz's(A city in Iran) dialect, to be specific. I've read before that no native Turkic words start with an 'r'. What is interesting to me is that in my dialect, or at least up until 60 or so years before in my dialect, no words contained an 'r' not proceeding a vowel. If we loaned a word from some other language we modified it to not have to deal with an 'r' proceeding a constant. Even native Turkic words are subjected to this. For example, "Rus" (Russian) becomes "Urus" in our dialect whereas Azerbaijanis from the Azerbaijani republic just say "Rus". The word "Kebrit" (A Persian loanword) becomes "Kirbit", "Reza" and "Rahim" (Arabic names) turn into "Irza" and "Irhım". The Turkic word "öyrənmək" becomes "örgəşmək" in our dialect and "Topraq" becomes "Torpaq". This has been fading however, as most of our population have learned Persian and can pronounce words that contain 'r's not proceeding a vowel. So my question is that did Turkic languages have a different 'r' sound? What about my dialect? Have we lost our own 'r' sound and got it replaced by the Persian one? Why couldn't we pronounce the 'r' sound if it didn't proceed a vowel?
These are really two separate questions, one about initial r-, one about syllable-final -r jumping across syllables.
The name Urum is derived from Rûm ("Rome"), the term for the Byzantine Empire in the Muslim world. The Ottoman Empire used it to describe non-Muslims within the empire. The initial vowel in Urum is prosthetic. Turkic languages originally did not have /ɾ/ in the word-initial position and so in borrowed words, it used to add a vowel before it.
This restriction was lost relatively early on, maybe even before the initial Turkish attack and invasion of Anatolia, due to the initial Persian and Arabic influence on the tribes in Central Asia, the Turkish assimilation of speakers of the indigenous languages of the region and the fundamentally Persian and Arabic veneer of the Turkish occupation and administration. Hence "Seljuk Sultanate of Rum" - anno 1077 - "Rum millet", "Rum" quarter of Jerusalem etc.
öyrənmək / örgəşmək - The initial syllable etymological root is really öğ (mind), not *ör.
topraq / torpaq or torba (in Tatar and Ottoman) - The speculated initial syllable etmological syllable root is top (ball), not *tor.
yaprak / yarpaq - The reconstructed Old Turkic word is *yapurgak.
This suggests no, Old Turkic was not like that, rather, metathesis is a tendency of local dialects, particular branches or newer forms of the language.
Metathesis is common globally, including in languages for which we do have thousands of years of writing, like Romance languages and Armenian.
For example թշնամի (tʿšnami), from *դշման (*dšman), which you will recognize.
Turkic languages like Azerbaijani Turkish rapidly expanded by assimilating speakers from the earlier indigenous languages of the regions in question - Iranic languages like Parthian and Old Azari and Kurdish, Georgian, Udi, Armenian, Greek, Slavic languages...
So it would not be surprising to see simplifications or simply other phonetic systems from those differing substrate languages.
The history of the expansion of Turkish in the original, Iranian, Azerbaijan is relatively distinct from that in Shirvan and the later Soviet Republic that took its name, the timelines, earlier civilisations and genetics are different.
Concretely, Iranian Azerbaijan, or ancient Atropatene, was the home of Median civilization. Deioces united the Median tribes in 678 BC and made the first Iranian Empire. Cyrus the Great was his direct descendant. The importance of the region for Iranian politics essentially has continued to the 20th and 21st century, both the shah and modern ayatollahs had roots in Iranian Azerbaijan, where the street language progressively became Turkish, and the Old Azeri language died out, but it survives in languages like Talysh and Tati.
Shirvan was home of the Udins or other Caucasian Albanian tribes. At the time of its Turkification, Iranian languages were spoken there - for example, Nizami of Ganja was a well-known Persian-language poet - but most people there at that time were still Udi or other Caucasian Albanian speakers and Apostolic Christians, similar to Armenians, as well as Lezgic speakers further North.