My second concern is that the assisted living facility has carpeted rooms and hallways so I want to choose the style of wheelchair and/or walker that will be easy for him to use. The new residence is unfurnished so my head is swimming with the cost of all the items he will need.
The hospital he is currently in has provided a wheelchair (quite small & tight fit) and the walker has tennis balls on two of the legs.
If you are furnishing your father's room, I would go through the place and see what others have done with their rooms to get ideas. Sometimes you see that things you thought he would need aren't that vital. Try to do it with a rep from the facility. They may have a chest of drawers or table they have on hand they would loan you or sell cheaply. Chat with the staff. It's amazing how sometimes things become available if someone knows you need it. Most of these places do provide beds though, although, if you want your own special mattress, Costco has the best quality for the best price.