Has a person fulfilled the obligation of lighting Chanukah candles if he uses a stolen menorah but owns the candles, match etc.?
This is not a complete answer but just my impression. I am assuming that the underlying concern is one of mitzvah haba b'aveira, a mitzvah whose performance comes about through the commission of a sin.
In the case of Channukah candles, there is no requirement to use ANY menorah http://www.dailyhalacha.com/m/halacha.aspx?id=2789
so if I can be mekayem without any menorah, I can't imagine that the menorah can be considered the object through which I effect my kiyyum of the mitzvah, so its status as stolen seems immaterial. The aveirah (I would think) can prevent my kiyyum but not undo it.