Including own published proofs in PhD thesis
Regarding copyright: It depends on the agreement you signed when the papers were accepted for publication. All of the agreements I've signed stated that it was OK to include the paper in a thesis.

Regarding self-plagiarism: Talk to your advisor, but typically you include a statement in your thesis that some of the contents are taken from your papers, and include a list of those papers. Then you can include your papers (in whole or in part) without needing to rewrite them. However, unless you're doing a "stapler" or "sandwich" thesis, you'll probably need to do some rewriting just to make the story flow.