How can I extract rusted/painted outdoor screws?
Apply heat.
A heat gun simultaneously softens paint and temporarily causes the metal in the screw to expand slightly. Once the paint starts to bubble and drip, your screwdriver should have no problem getting properly slotted. Start with a minute or two of heating on the low setting and test with a screwdriver. If that doesn't work, try again with higher heat. Unless the head was already stripped, you should have no problem getting your screwdriver engaged. As the screw cools, it should also loosen up and be easier to remove.
Heat guns are inexpensive and useful for a variety of projects. Before I bought a heat gun, I used the following steps:

Remove the paint/rust from the screw head.
For paint, I've found a gel paint stripper to be effective.  As you point out, naval jelly or a penetrating oil can be used on rust.  You can also try using a rotary tool fitted with an abrasive brush to quickly clean the screw head.

Use a manual screwdriver to get the screw started.
Too often, I've gone straight to the power screwdriver and stripped the slots in the screw head.  This is particularly common with Phillips head screws.  Using a hand screwdriver reduces the risk.

Re-slot the screw head.
Using a cutting disk attachment on a rotary tool, it's possible to either add a slot or deepen an existing slot on the screw head.  Aim to create a single groove deep enough for a flat head screwdriver to engage.

Drill or chisel out the surrounding area.
If all else fails, you ought to be able to extract a screw by drilling out or attacking the area around the screw with a chisel.  (Presumably you could use a rotary tool for this step as well, but since I obtained one I haven't need to go past step #3.)  If you dig out enough material, you might be able to grasp the screw head with a pair of pliers.


It probably goes without saying (but I will anyway) that you should wear proper safety equipment such as gloves and goggles when handling harsh chemicals and using high-speed rotary tools.