Yes, you can replant the button and it will grow a taproot, the taproot will heal and regrow a button. Yes it takes a long long time (many years) and probably isn't worth the effort because you'd get growth from the original cactus the whole time the 2 sections would need to heal.

Yes, harvesting loph's without killing em is simply a matter of making sure to leave the taproot in tact and leaving a bit of the green flesh above to regrow when you cut a button. There are guides on erowid and elsewhere online if you google "peyote harvesting".

I do believe you can cut and root pups from a caespitosa but again it takes a long time for a cut loph to re-root