What actually happens to my messages if I uninstall Messenger?

Nothing is lost. Your messages live on Facebook's servers tied to your account, not on the app. Uninstall, reinstall next week or next year, log back in and every conversation is right there. Uninstalling only removes the app, not the data.

Fully closing and reopening the app (swipe it away) fixes more than you would think.

Safe to delete. The only thing you lose is local cache and any draft you had open. Log back in anytime and your history syncs down again.