According to the Bible of French conjugation, it’s a mistake.

It’s a common usage according to the google.fr strict search with “vous fairez”.

As a native speaker, I can explain that by the ambiguity of the verbs of the “third group”, a lot of people make a lot of mistakes every time…

As the examples you turned up show, fairez was not too uncommon in an earlier era of French; see this Ngram chart. (Unfortunately, the latter half is misleading because when you check those examples, they’re books from much earlier than the late 20th century, despite Google’s metadata.)

Your first two books don’t actually contain both different spellings, just fairez. In the first one, the ferez you seem to see are in fact serez with what was then the cursive s (check the other examples of s in that text). I assume the second example results from poor OCR; serez is the only possible reading in those cases (notice that they parallel fut, which is also a form of être).

At the time these were written, fairez was comfortable in the company of words like seray, recepveur, voicy, and this charmingly spelled phrase I turned up: « une beste de vostre troupeau » .

Some of these older spellings are mere inconsistencies of practice, typical of any written language before spelling reforms and standardization. However, others are consistent but represent an earlier pronunciation. (The s in vostre is a good example.)

The appearance of ai in fairez, as @jlliagre suggests, would not be in any way unexpected; after all, the word is formed on the stem of faire. In fact, modern spelling is inconsistent because some of the vowels that are now reduced are written e but others are still written ai in faire and other verbs. The crystallization of irregularities for posterity is the hallmark of language reformer and purists alike.

We certainly need to know when the spelling was reformed, and why (how long before had the pronunciation shifted?), to analyze Dostoevsky’s example. It requires more research than I have the time or inclination for right now, though I do notice a spike in fairez in the late 19th century in that Ngram chart. However, among the examples I did look at, even some of those 19th-century ones were misdated for, or quoted from, books in the 16th and 17th centuries… Hence, your guess that Dostoevsky’s Russian milieu was working with outdated data may be reasonable. As you probably know, the 19th century was a time when Russia was just beginning to open up to Western European influence, and an influx of classic literature (e.g. Chekhov has characters quoting Hamlet in The Seagull) might well have resulted in older forms being mistaken for current in their secondary literary home at the time. Perhaps Dostoevsky was even aware that it was archaic and used it for effect.

My apologies for not being able to do more than speculate on that last point at the moment! If I have time to research more later then I’ll update this answer — or if anyone else cares to share what they know I’ll gladly incorporate it.