1. Not in my experience.
2. If it interferes with the proper closure of the jaws (rust on mating surfaces) then it certainly will affect the measurement (current will read low.)
3. I have no idea, but I would try abrasive (other than emery which sheds conductive powder) rather than a chemical rust remover like Naval Jelly. If the latter somehow got into the electronics you would be in big trouble.
The visible part of the jaws on my clamp on meters are not shiny, but they work fine. When I think of rust I think of something you can scrape off and have a pile of crud be it big or tiny and there are voids left on the remaining metal.
What kind of problem are we dealing with here? Lack of shininess or some major corrosion?
How have you been storing the clamp? If moisture is the problem I think the rust spots will be the least of your worries. Depending on the accuracy you require I’d have it checked out for internal corrosion.