Is it true that most linguists assert Basque has not substantially changed?

Not to my knowledge. The late Larry Trask, preeminent vasconist of his time, spent a lot of leisure time refuting inane claims about Basque being related to every language on earth, and part of his armoury as a historical linguist was that such inane efforts made use of modern Basque dictionaries, whereas​ both what we have reconstructed of proto-Basque, and what little we know of Aquitanian, the ancestor of Basque, are phonologically different.

How did the Turkification of Byzantine empire take place?

I’m wondering how the old Greeko-roman culture of Byzantine, perhaps the most sophisticated in the middle ages, just collapsed and replaced by language and culture of tribal immigrant Turks who hardly had any written tradition?

For the specifics of what happened in Anatolia, see When and how did modern Turkish become the majority in Anatolia? (where I’m about to post an answer).

For the general principle: people don’t adopt a different language and culture because they’re choosing between Cyril of Alexandria and Avicenna, or between Michael Psellos and Ibn Tufail. People adopt a different language and culture because it will bring them direct benefit in their daily life, in the regime they happen to find themselves. And we’re talking a lot of peasants in Anatolia, who had never heard of Psellos or Cyril.

Are there reverse Latin and/or ancient Greek etymological dictionaries?

At a stretch, you could use Pokorny’s Indo-European dictionary to move forward, although it won’t move terribly much forward: you’d have to do the work of getting from Old English to Modern English yourself.

I’m not aware of such dictionaries myself, though I’d be surprised if someone hasn’t done one. These days, your best bet would be the text of an English Dictionary that allowed its etymologies to be searchable. Or, as Lotte Meester has pointed out, Wiktionary.

Are the many “i”-like combinations in modern Greek comparable to the “yat” and many “i”-sounding letters in old Russian orthography?

There is one major similarity between the Old Cyrillic and Greek alphabets: originally, both were (mostly) phonemic, but several of the distinct sounds represented by different letters merged later on, so that there was two or more ways of representing the same phoneme with different letters. So the letter Yat seems to have originally represented /æ/; its sound merged with /e/ in Russian, so that /e/ was written as both <Ѣ> and <е>.

In the same way, Greek for instance, used to have different pronunciations for <ω> and <ο>, /ɔː/ and /o/; Greek lost its quantity and quality distinctions, so to this day there are two ways of writing /o/: <ω> and <ο>.

On the other hand, the two different ways of writing /i/ in Old Cyrillic, <и> and <і>, were not comparable to Greek—they were inherited from Greek, where <η> and <ι> had already merged in pronunciation. Old Russian orthography had worked out rules for when to use one and when to use the other (Dotted I (Cyrillic) – Wikipedia); but those rules were artificial:

In the early Cyrillic alphabet, there was little or no distinction between the Cyrillic letter i (И и), derived from the Greek letter eta, and the soft-dotted letter i. They both remained in the alphabetical repertoire since they represented different numbers in the Cyrillic numeral system, eight and ten, respectively. They are, therefore, sometimes referred to as octal I and decimal I.