The floor: I'm not sure about this but if it really isn't rotting you could tear off the peeling bits and fill the torn spots with floor leveling compound. I'd go with a vinyl flooring over top.
The ceiling: I'm thinking marine drywall and vapour barrier, and maybe some insulation above.
Ventilation: Does this room have ventilation to the outside? You need it to remove the moisture that a shower leaves in the room, or you're looking at more damage. If not, a fan would be good.
Brrr: Is it cold down there? There are these great little spot heat lamps that you can install in the ceiling to take the chill off of a selected area.