2. If you can get to the back side of the hole, you can fill it with a marble-sized lump of 2-part putty epoxy, such as QuikSteel. Fill the stripped hole and the channel behind it, let it cure, then drill it to the correct size. J-B Weld may not work, as it is very runny.
3. For a fast fix, just try a bigger screw; but if the edges of the hole have cracks (stress fractures), the aluminum will just strip again, so drill a clean new hole first, and use a tap to cut clean threads. I would only do this if the metal is fairly thick.
4. I've heard that aluminum brazing rods work well and are fast, easy, and cheap. You can pick up a few rods and a MAPP gas or propane torch in the welding section of most hardware stores. There's a Welding Tips and Tricks website that compares three types of aluminum brazing rod (HTS-2000, DuraFix, and Aladdin 3-in-1), and includes some instructional videos. However, the heat of the torch will probably blister the paint on the door.
5. The hole could be MIG or TIG welded, but I doubt many welders would do a job that small.
6. Heli-coil inserts only work if the metal is thick enough, but it isn't on a screen door.
Hope that helps!