....hmmm, there are mermaids.
NO! NO TAMAO ICHINOSE SOCIAL LINK FOR YOU!
He can punch an average monster to death even as a regular human.
Depends on the monster.
Mermaid? Yep.
Lizardman? Yep, although it might get a little dicey if they gang up on you.
Witch? Depends on the experience/abilities of the individual in question, but speaking generally, if you can close in and take away their fancy stick before they cause trouble with it, you're golden.
Snow woman? If she doesn't freeze you solid quickly enough, nor rend you with her ice claws, yeah, you could beat one up.
Succubus? Eh, this one comes down to luck and planning. Succubi can fly really fast and are very agile, not to mention having nails sharp enough to cleave big thick trees even better a hot knife would butter. If you could survive long enough to get her into punching range, though, yeah, you could kick her ass.
Werewolf? If the full moon is shining on them, you're dog food. Otherwise, you stand a decent enough shot at winning.
Catgirl Ms. Nekonome? Easy-peasy.
Kraken? Again, if you can get into face-punching range quickly enough, you should be good. But if he grapples with you before you can (which is pretty damn likely, since, y'know, tentacles and all), well, you're definitely not good.
Lamia? As long as she can't brainwash you, yep. Although her tail could probably bash you around a bit.
Transformed Saizou? A critical hit to the head might take him out if you're lucky, but if he can hit you first? You're dead.
Troll? Nope, their skin is too tough even for your Jotaro-tastic punches to have an effect, and you aren't taking any hits from them either.
Kitsune? You're cooked, man. Itadakimasu~!
Vampire Inner Moka?
KNOW YOUR PLACE, HUMAN!
The Pillarman Headmaster? Hah, you're so fucking dead without Ripple and/or Star Platinum.
2. Pretty sure that library isn't going to go anywhere anytime soon, while Joseph could go off on some evil-slaying business trip and leave us with no way of contacting him.
For all Jotaro knows, the entire Academy could be a creature of chaos, constantly moving around the rooms that are inside it from one day to the next, a la Castlevania on crack. .... it isn't, but it could be.