Does the hotel have handicap accessible rooms?

No

They have a handicap accessible door on the first floor for first floor access. No access for the upper levels including hotel rooms.

There are handicapped accessible restrooms in the lobby

There are steps to get in the building. I don't believe it has a wheelchair entrance

Also, as far as I know, there's no elevator up to the rooms.

I agree; they do not.

No, don't think so, building is old enough to be grandfathered in without it.