Were Imperial Russians aware that their greatest enemy at the time, the Ottoman Turks, shared cultural and genetic ties with the Mongols who conquered them centuries prior?

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I admit that my knowledge of Imperial era political standings is pretty limited, but as far as I'm aware, it seems to be very unlikely. Genetic ties are a tricky thing even today, let alone in 18th century. And culturally Turks were obviously very different from steppe conquerors of old. Not to mention that it's been ages since Mongol Empire vassalized Russian principalities, and it was more of a distant memory who exactly these conquerors were and what other ethnic groups may be similar to them. Since 14th century Russians actually considered The Golden Horde (and later - Crimean Khanate) to be descendants of their former conquerors, since they more or less were reminiscent of Mongols and inherited both culture (albeit changed in the coming centuries), political position and attitude towards their neighbours. I don't recall any instances of connections being made between Ottoman Empire and Mongols, while Tatars were ever present and familiar threat.

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Did they think of the wars they waged as payback against their former invaders?

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Such sentiments were probably present during the time Russia tried to get rid of the Tatar Yoke, but later Russian state would actually actively vassalize and enlist former Mongol Empire remnants, considering them to be not only enemies, but also allies. By the time 18th century came it's very unlikely that Russia as a whole or Russian rulers as persons deemed Turkey to be a symbol of the past tensions between Russia and the steppe. Conflicts with Ottomans since late 16th century from Russian perspective were usually considered to be defensive wars or actions to enfore their will in the Mediterranean and Balkans.