Yikes.
A lot of reinforcing of the behavior you DON'T want to reinforce, like making a picky eater.
I would start by getting this dog comfortable walking up and down the stairs. You're lucky you can physically carry a dog down the stairs, but it's likely a very unpleasant experience for him, and if I had a choice between being carried down the stairs or peeing in the house...I think I know which one I'd chose.
I know how difficult it can be, believe me. It took me weeks to teach my first greyhound how to use the stairs at my old building. I could NOT lift him, so for me it was moving each paw up each step (three flights) over and over and over. Eventually he just...walked down the stairs! Problem solved, but it wasn't easy.
I feel like if he's not afraid to go down the stairs, the other matters will, one by one, start to resolve.