You have said it is an A500, but if it happens to actually be an A500+ (it should say so on the printed label, if there is one), you may have a leaky battery problem to deal with. The plus models had an internal real time clock that ran on a NiCd battery which almost certainly has leaked corrosive electrolyte after all this time. Depending on which way the computer has been stored, such a leak might cause issues with the mouse/joystick ports or anything else.
Even if it is not an A500+, you could still have a look on the inside of the computer by opening the six or so screws on the bottom and carefully lifting the top half of the case off. It might be that some of the socketed chips are not completely pushed in, though usually this causes the machine to not boot properly at all, which in this case it seems to be doing fine.
Of course you should clean the mouse and do all the other external checking first, as others have suggested.
