Great space! Yes, backyardtheater.com for your equipment questions.
You can get direct-bury electric cable, but I would bury a conduit and run a 20 amp line out there. Same with the coax. I would keep the coax at least a foot away from the electric line.
Personally, I wouldn't leave the projector out there. I would build a shelf mount and run all of the necessary cabling to the shelf and hook up the projector when you are going to watch a movie. If you get a new projector, the only hookup should be plugging it in and connecting an HDMI cable. Even an older one like mine only requires connecting the three component cables (R,G,B). Once you do it two or three times it won't take more than 5 minutes to setup and take down.
You can build (or buy) a small cabinet to house your DVD/BR player and amp (and any other equipment) and just close and lock the door.
Amps are pretty cheap.....projectors not so much.
Look at monoprice.com for great prices on speaker wire and long-run cables, particularly long HDMI cables.