When I adopted Desmond, the shelter people warned me that it could take several days or even a few weeks before he felt safe enough to move freely around the house. So I would not be concerned that Felix is hiding under the bed. I would put his litter box, food and water in the bedroom and even shut the door, until he starts coming out on his own. Maybe go in and talk softly to him from time to time, w/out trying to pull him out, so that he gets used to your scent and the sound of your voice(s). And yes, keep the dog away from him for now.
This is what I was prepared to do, with Desmond. Desmond, however, had other ideas.