Permiting Questions: Will they report non-related permitting violations?
It's hard to say, so you'll have to let us know how things like that work in LA, but it seems likely that it will come up. Regardless, I would approach it headon, i.e. properly, and see what happens. Particularly with structural changes and electrical work involved, you most likely are only making it worse for yourself in the long run by prooceeding without a permit simply to avoid issues with the shed. And, paricularly since you mentioned setback possibly being an issue, the matter of the shed is apt to come on its' own at some point anyway. While someone driving by can't necessarily tell if you pulled a permit, they can surely see it, and have some idea as to the approximate setback. As long as you're going to be working on the garage anyway, maybe it would be good to just connect a shed to the garage, if that would put you in compliance.