Not sure how many latrines there should be, but some of the larger castles had their own latrine towers. Also keep in mind a lot of people used chamberpots.
As for washing/bathing facilities: in Medieval times people mostly used bathtubs. The bathtub could be placed in a room (near a fire for example) or outside if it was good weather. Rooms set aside for bathing were pretty rare in castles I think, but it did happen.
Finally, the laundry. Keep in mind that people back then had less laundry. Clothes were very expensive and most of the time the outher clothing was never washed (it was brushed). The laundry in castles was done outside or in a designated room by laundresses.