Unless you consider Ace Combat an FPS, the 3DS doesn’t really have any of them that I know of. You’d be better off playing the old DS ones. If you count Homebrew, your options expand a lot. Doom and Quake play wonderfully on the New3DS with the C nub.
As far as my favorite games on it: wow, there are so many. Shin Megami Tensei IV (and Apocalypse), Fire Emblem Birthright, Animal Crossing New Leaf, Pokémon, Monster Hunter 4U. The list goes on.