Yes, you can. If you have access to re-categorize threads (from the pencil icon next to the title on the original post), you are essentially just changing the topic-header. I just moved this thread from Site Feedback to General and it retained the two likes it had.
If you do it a different way (copy/paste, quote text) where you’re pulling text out from its original header, it won’t retain thread information.
Unfortunately moving posts piecemeal is a Moderator and above option only. Is it just certain messages? Point me to to them and tell me where they go. (I have about 15 min before work, so be patient if I don’t get this till later! )
Also…I think when you start tearing threads apart is when your likes and clicked link data might go away or get screwy, so this may be going around the world to reach our backyard…I imagine the likes will stay with it.
Could you screen shot your posts or bookmark them?
I don’t remember what level, but you might be able to build a poll from the gear Icon that might be a better way to have people vote rather than track likes. Might be too after-the-fact, but I believe that option is open to you.