For your chattering flipper, you might check the top of the coil, where there are two very fine wires going from the coil lugs to the body of the coil. If the fine wire on the hold side breaks, you would have a problem very much like you are describing. If it is loose and moving, all the more likely that is what is going on. AS for the bolts, I have used PB Blaster or similar to loosen up the rust, but you need to wait a good long time and leave the game open for a while prior to turning it on if you use this type of product due to risk of fire. My advise is to check the fine wire as described above, which can be soldered if it has broken loose prior to going to a lot of trouble to get the coil off and replace it. Hope that helps.
Ron