How can I remove surface rust from a cleaver and keep it off
For the rust you can try an oxalic acid cleaner like Barkeeper's Friend (try it on a small spot first!).  It's worked great for my bakeware's cooked-on spots, and should be useful for rust as well.  Using it will be a bit of trial-and-error, because leaving the metal exposed for too long will etch it, and not long enough won't clean much.

For rust prevention, just clean the cleaver after each use and thoroughly dry it.  I'm not sure what it's made of, but those steps should suffice to prevent rust formation.  If you're obsessive about this, or don't cleave often, you can coat it in petroleum jelly afterwards (I do this with my straight razor.  It's probably not feasible for a kitchen tool unless you're storing it away).