I've done a shut down on a dog like this, and it worked very, very well. I don't do it with every foster, and I was skeptical about the shut down after reading all the debates here on it --but I decided to try it with a shy, withdrawn foster and just see for myself if there was a benefit. There was, in just a week.
The reason it might help here is by going under the table, the dog is seeking a den, shelter, a safe place to watch where he won't feel vulnerable. A crate would give him that. It would let him observe your routine, get used to the smells and sounds, and decided you aren't threatening and that the whole house is used by his pack, on his own time. It may only take a week (mine did). When he's out, keep him leashed next to you, even in the house so that he learns all good things come from you. Feed in the crate.
Stairs should wait for trust. Many, many dogs find these super-scary at first. Once he trusts more, you can put high-value treats on the stairs and coax him up slowly. It may take a few days -- he may only go up three stairs the first day then freeze (hooray! three stairs), but four the next, etc. Same goes with down the stairs, which is WAY scarier for many dogs. I stay below and next to the dog going down, so they see me ready to catch them, not the long drop down.
We trained a BLIND dog who was afraid of stairs how to do this -- you can do it with a sighted puppy easily. Just be very, very patient and be prepared to go through lots and lots of yummy treats while you build up his courage.