Why does Jason bother to cover his face at all? He used a potato sack (or whatever it was) in the first couple of films, so it clearly was not for Rule of Cool (not that a maybe undead, psychopathic murderer would/should care about that anyway). His sole purpose in life seems to be to terrorize and/or kill anyone 13 or older that he comes across— his hideous mug would be great for doing the former (as seen in Jason Takes Manhattan). There's no chance he's THAT sensitive about his physical deformities, is there?
Looking for anything resembling genuine characterization in the series is a mostly lost cause, but there are occasional signs throughout the series that at the end of the day, Jason Voorhees is an angry, cruel ten-year-old boy in the body of a seven-foot-tall immortal serial killer. He's actually reasonably sensitive about his disfigurement, to the point where he may wear the mask so he doesn't have to look at himself.
And "Jason Takes Manhattan" has him show he knows how ugly his disfigured face is, considering he lifts his mask to scare away some guys bothering him because he didn't have the time to kill them.
Yes he is that insecure about his looks. It's a hold over from when he was alive and whatever caused his mental handicaps also left him deformed as a human and children can be cruel. In addition everything we've seen about his origins are at all true the camp counselors were typical teenage pricks and probably gave him more than his fair share of shit.
Where does Jason keep getting new outfits? In many of the movies (especially the first few) Jason has a completely different set of clothes to the last one. Does he go shopping or something?
Out of Verse because keeping realistic clothing damage between multiple films would be nearly impossible, it's probably supposed to be the same outfit and the props department didn't give a shit. It's hard enough to keep things consistent in one movie. Inverse Jason is larger than your average male but not monstrous. It's not difficult to believe that he's able to scavenge enough clothing from the two parks (Crystal Lake and whatever the neighboring Camp from Part III) and the surrounding city. It's not entirely clear how far Jason's territory extends and he only seems to change between flicks which other than the first few films are implied to be years apart, usually long enough for the notoriety of Jason to die down some. A few years is more than long enough to grab one or two outfits and Jason is shown to be clever enough when he needs to be.