As far as how often to clean your pickups, I'd recommend just doing it whenever you take the occasion to change strings, perhaps once every three months or so.

You could remove the screws, soak them in vinegar for a day or two, and then use a wire brush to scrape off the rust from the screw heads. The pickup pole pieces, you could try a little toothpaste on an old tooth brush, or use a tiny bit of metal polish with a Q-tip.

As a rule of thumb: never spray anything directly on the pickups. Gently vacuum the pickups, clean with a damp cloth, apply a small amount of polish on a cloth and apply on the pickups if necessary, use a soft brush to remove excess and a dry cloth in the end.