You cannot change that from non self-moderated to self-moderated thread or vice versa after you published the thread.
You're half correct: you can remove self-moderation status by (temporarily) moving the thread to a board that doesn't support it.
Create a new thread with the same content may never be appropriate while the previous thread is still active. If I create a new thread with the same content, I'm afraid I'll get repeated post in the thread which could be categorized as spam from many users.
I'd make a post with a link to the new thread, and just lock it.
Like I did here. You could even move the old thread to Archival
it will even be wrong for it to later be self-moderated. Some people wouldn't have posted on the thread if seen as self-moderated, and you can not now deny those people of their rights.
Correct, so it won't be made self-moderated.