Use a penetrating oil such as WD-40 to dissolve the rust. Add a few drops at the screw head and let it drip down the shaft. Wait 10 to 15 minutes for the oil to work, although it may require a few hours or even overnight to dissolve heavy rust.
Heat is one way to remove paint from a screw head -- but use caution around flames and paint. Hold a large nail with pliers and use the flame from a lighter or match to heat the nail head. Press the nail head on the screw head to melt the paint. The screw will expand, so use ice to cool and contract it before removing it.
Screws stuck at an angle can often be coaxed out by placing a flathead screwdriver at the end of one slot and lightly tapping the screw counterclockwise.
You will need to make a new head for a stripped screw. Use a file or roto-tool to carve a groove in the screw head, then use a strong flathead screwdriver to remove it.
When the head has been knocked off a screw, you may decide it is not worth the effort to extract it and decide instead to nail it in and caulk over it. If the screw must come out, use needlenose pliers to turn the screw out.
This option requires a special tool and drilling a pilot hole in the stuck screw. Insert a screw extractor that is slightly smaller in diameter than the stuck screw and turn it counterclockwise until it engages and removes the screw. Screw extractors are available from tool corrals in hardware stores and home improvement stores and are a must for anyone doing a lot of carpentry work.
You may need to drill out a stuck screw using an electric drill. Use a drill bit the same size as the screw and aim straight down the middle of the screw. This will not work if the screws are stainless steel or if a screw extractor got stuck. Wear protective goggles when using a drill.
When working with screws, pass over bargain-priced screwdrivers and choose a strong, professional quality one. This is especially important for Phillips head screws. Use the right size screwdriver for the screw and, to prevent stripping, do not push too hard on the screw head. A tip from the pros: Dip the screwdriver bit in valve grinding compound for a better grip with the screw head.