There's a solution for dealing with wood screws which have broken off below the surface of the wood. If you don't want to risk damaging the wood by digging it out, try driving the screw deeper into the wood with a nail set. Then just fill the hole with a wood filler, and drive a new screw next to the broken one.
Weather stripping can seal leaks around movable joints, such as windows or doors. You need to choose a type of weather stripping that will withstand the friction, weather, temperature changes, and wear and tear associated with its location.