Once in a while when landing bigger largemouth my finger(s) will break through the thin flap of skin under the bass’ chin. Does that heal eventually or does it stay like that?  

Based on the one detailed study of hook wounds in bass I'm aware of, it will eventually heal, usually within 2-3 weeks, but depending on the size of the hole, could affect feeding for a while while it does. Should be a rarity though and not happen often unless there was already some type of small hole from a lure or a cull clip. Might double check how you're gripping them if it's happening more than a time or two.

I bet it will heal, bass are pretty tough, that doesn’t mean that it’s a good thing thought, try adjusting your grip slightly. 

I caught a small largemouth on my last fishing trip, it was hooked in the top part of the mouth. As careful as I was unhooking it I tore the top part of its mouth...really bad. I felt so bad it almost made me sick! It swam off but it just broke my heart to mess up that fish. I pray it makes it. 

As long as the bones arent broken it should heal eventually.  Things happen, just try to minimize damage to the fish as much as possible.  WHat dri ves me nuts is catching a 2lb fish and seeing where some googan cut a 3" slit in the bottom of his mouth to weigh him with some crappy wal mart scale.