I'm not saying STICK to Game of Thrones, but the point here was to make "Russian Game of Thrones", not just another Russian fantasy setting. The monsters are okay and all, they just need to keep faithful to the tone, etc.
So zmey, for example, can be the real masterminds behind Valyria instead of serving the humans, or they could be just legends (maybe real dragons are unintelligent), or maybe they're degenerate (like it used to be zmey were intelligent and multiheaded, but with magic on the wane they became regular old dragons).
(also, just to note, you're way exaggerating the prevalence of multiple headed dragons in Slavic folklore. It's mostly that the most iconic of them, Gorynich, is three headed. There are plenty references to single headed ones all over Eastern Europe and Russia).
Giants - probably less monkey men than ASOIAF, giants in Russian folklore are noble and sophisticated. They just can't help causing destruction due to being so big.
Various water spirits would be just the right tone for ASOIAF if you only accentuate how scary they are. Rusalka, vilya, stuff like that. All about drowning people.
Leshy are already plenty ASOIAF as it is, what with being hideous moss covered giants addicted to raping, drinking and gambling.
Domovoi are good. Just make them less magical.
In general, so long as you tone down the obvious magic and make things more like "animals that you could sort of kinda imagine possibly evolving naturally somehow", it fits. On top of that you put magic.
I'm also in favor of Phillip Pullman style witches ruling the taiga, but I say that about every setting.