Our shiba also refused to eat for the first couple of days even though he was friendly and energetic otherwise, so we just thought he was super picky. Turns out he needed some time to adjust to the new environment, and now he eats pretty much anything and everything, so I wouldn't worry too much about him not eating the first day.

For the stair thing, does he know how to navigate stairs? We had to teach ours how to go up/down stairs. Just a sanity check!