Proxima Centauri is a rather pathetic red dwarf. I wouldn't count on anything habitable around it.
Astronomers used to think this as well, but recently we have been finding out that habitability is very complex, and a few studies have shown that we can't rule out the existence of habitable planets around M-dwarf stars afterall.

You're correct about the habitable zones of α Cen A and B (going with the original and somewhat dated definition of the term). There would be one of these zones around each of the two stars, of about the same size as Sol's, and orbits within these zones would be stable over million+ year periods.