My rust remover is 0000 steel wool and some penetrating oil. You can let frozen screws soak in it as they often get stuck in the bridge hardware. Clean all the parts with soap & water and fully dry them when you're done or your guitar will stink
What do you mean the coating has worn off the frets? Frets are all but always made of nickle and have no coating. The only coating they should have is a dull tarnish or they may even turn green. If you polish the fretboard like you're supposed to with lemon oil & a rag and apply some elbow grease to the frets in the process they'll shine up like new.
If you use the steel wool on the pickups without protection then you will dull up the plastic. Cover the plastic with masking/painters tape and have compressed air on hand to blow off the wool fibers that are going to stick to the poles.
If would be cheaper and easier to take the guitar to a pro for the nut. It takes precision tools ($$$) and a trained hand to do properly.
