Many libraries still do -- I don't know if it varies by geographical area or just by individual library. Depending on how much effort you want to put into it -- I realize you don't have a ton of time -- it could be worth checking a couple of other nearby libraries (or googling "friends of the library," which is often the name of the org that actually sells them, + a few nearby locations).
Yeah, they could. They're businesses too -- they're not necessarily going to spend money on storage or disposal for product they can't move. I'm sure it varies case by case. I had to shrink my library ~10 years ago and I remember the buyer being very picky about what they'd take. (Also, if you go that route, do try to sell them first, don't feel you have to start by offering to gift them. You might get enough $ for, oh, a pizza
IIRC my process was (1) library book sale table, (2) used book store, (3) homeless shelter. Goodwill/Salvation Army seem like good bets too -- better bets, these days, than the ever-more-rare used bookstore.