Message Contacts aren't syncing with Contact Book Contacts

In that case, the contact has been stored on your phone. It's possible that it's in a different group or account. Go to Contacts and tap "Groups" in the upper left corner. Make sure that all the options are checked.

If you see a contact in Messages but not in Contacts it's because you never saved it to Contacts. Open the conversation with that person. At the very top of the screen, tap on the little head icon. Tap "Info". Then, tap the little arrow at the far right of the line with the number on it. You should then be on the screen where you can create a new contact or save it to an existing contact.

I don't know the answer to that question. This article presents a method that looks as if it should work, though:

https://michaelkummer.com/tech/apple/move-iphone-contacts/