I don't know whether the Message_ID survives being moved to a different thread, though it seems likely. The IDs of deleted messages are not reused. At least, not by design - if the system fails and is imperfectly restored, then any 'permalinks' recently posted - maybe to other forums, might be wrong.

And the permanence of links to Mudcat depends on how the Mudcat server redirects old ones. There used to be several sub-domains, as I described this post, so old posts - not necessarily on this forum - may link to, say, http://awe.mudcat.org. For that link to still work the mudcat.org server needs to redirect everything - which AFAIK it does - or did. I posted links to Mudcat on usenet years before I ever posted here. Do they still work?

I can't find a very old usenet post linking to Mudcat - but here, for your amusement, is a selection of posts from 20 years ago.
https://usenetarchives.com/threads.php?id=uk.music.folk&y=2005