Air sprayers are FAR superior to the electric (Wagner) type!!!
Sounds like you have NO experience with spray painting. Therefore, you could choose option 2 and paint them with a brush.
Before spraying the doors, get some practice. You need to practice and adjust your spray pattern typically using old newspaper. Once you have the spray pattern adjusted right, try spraying a large piece of cardboard. Get used to moving the hand & sprayer first then pulling the trigger just before the beginning of the cardboard, also releasing the trigger at the other end of the cardboard. This will also help you identify whether your air tank has enough reserve capacity to do the job right. After some practice, you may feel you are ready to paint the real thing.
Remember not to spray too thick. Use multiple coats. Might get a book on spraying to read up on techniques.
For your other question, I would always use oil-based for exterior. However, remember you cannot paint oil-based over latex. You can spray any liquid through a sprayer.