I'm no carpenter but I'd have to agree that building your own shed at those dimensions would be better AND it would be custom. The trouble would be pitching the roof if you were inclined to do so(no pun intended). You'd wanna go with a plastic or metal roof anyways because your eaves would need to breath if wood or your roof would rot. With a plastic or metal roof you could insulate the roof and not worry about rot.

If you do build one it is easy enough to insulate it with paperback fiberglass batting. Then just get some nice cheap 1/4" ply or wallboard and finish the walls. The temps can be dealt with, but it would take either A. alot of work or B. decent chunk of $ for a decent temp controller.

I'm also no electrician but there is ROMEX out there appropriate for outdoors and you could bury it to your new shed. Splice into an existing circuit, though I wouldn't recommend doing it unless you either have decent AC experience or get explicit instructions from someone qualified to do it. (May be harmful/fatal to yourself or others if not done correctly.)

Also there's what Stinky said, much to worry about. I don't know how your property is set up but there's nosey neighbors and possible intrusion/dectection would be your biggest concerns after all that. I'd stick to your house if I were you, but if you want a shed then it's definately doable just risky.