Definitely NOT an Indo-European language and possibly no linguistic relatives (at least any major language relatives). It has its own, unique alphabet, as does Armenian, but both were ultimately derived from the Greek alphabet.
Yep, Hungarian & Finnish are related as is Estonian to them. Estonian & Finnish are a bit like Spanish & Portuguese with Hungarian a lot more distant. Say like Spanish & German or something.
I would say that the "distant-but-still-close" to portuguese/spanish is romenian.
Armenian is indo-european btw, I think you know that, just said to avoid anyone thinking its not.
The big family of hungarian, finnish and estonian is the Finno-Ugric family. There is some polemic hypothesis that Finno-Ugric and Indo-European families are related, and I wouldnt doubt it given the proximity of cultures, and some very strong resemblence on the pronouns.