We did this a few years ago in a house that was built in 1945 and the answer we heard most often was, "Don't disturb the flooring, just cover it" So we used floor leveler to fill the pattern in the flooring and laid the peel and stick tiles on top of that.
On disposal of the old floor, if you do take it up. My father-in-law works with Habitat for Humanity and they just double bag it and toss it in the trash.
I would just double bag it and put it in with your regular trash. My local landfill wont take drywall, but if its in a bag, they take it.
Kinda silly, cause now they wont take 'fridges unless the freon is all gone, but dont have any way to reclaim the gas at the dump, so everybody just cuts the pipes and lets the freon out before you get to the dump. Real smart.