Randomly choose an old post and reset the publication date to now. Promote older posts back to front page and RSS feed.
No. The original post has the publication timestamp changed. There remains only only copy of the post.
Yes it will. Did you read the warning? This plugin should no be used with permalink structures that include dates. This is yet another reason why you should not use such a structure.
Glad you asked... Remove Date from Permalink Structure
I am working on several others, when released you will find them on my WordPress Plugins page.