Do you mean: (1) I am a friend of Tom. (2) I am a friend of Tom's. If you do, then, yes, the double possessive is fine. It is very idiomatic (the way native speakers use their language). Maybe in very formal writing, however, it might be better to use sentence #1. But in regular conversation, #2 seems more natural to many (most?) Americans.

I recommend re-posting your question under, "learning Russian." You will be more likely to receive a response. :)

Correct: 　Таны ах эмч үү? 　Чиний эгч эмч үү? 　Ах тань эмч үү? 　Эгч чинь эмч үү? Таны ах тань эмч үү? - sounds strange. Definitely not formal, but you might hear it in daily conversation...