Hmm, I'd probably have to go with the Half Life series for best single player, TF2 and Unreal Tournament for best multiplayer. And Prey gets a mention because it was pretty awesome.
It is a 'full on' genre, just not a lot of games are First Person Puzzle games. A lot of games have incoporations of it (like Half Life and Metroid) but they aren't just puzzles. Something like Portal or the UT3 mod 'The Ball' are First Person Puzzle games.
Technically, as you said it. It's usually referred to as point and click because I don't think you have ful control over your character movement. Narbacular Drop is another first person puzzle game, it was made before Portal, the guys who made Portal made this game, Valve hired them for this game. :>
Fallout 2 is not a FPS not sure if you can really count Fallout 3 as one either. Anywoo, I don't play many FPS mainly as I suck at them, but one that I used to play the hell out of is Quake IV and Doom 3 and for some reason F.E.A.R intrigued me for a bit, but I still suck at them all. LOL
Fallout 3 can be approached as turn based. Take your turn selecting body parts to shoot and then the enemy takes its turn while your AP recharges. Turn based rpg, Fallout 3.