Well in The Measure of a man Data
opened up drawers to get stuff out. Perhaps each room has book shelves and places for nick-knacks, pictures and books, but they have a sliding door that covers them by default. That would keep a lot of stuff in ones quarters out of sight and safe while in battle. If those compartments are open, once Red Alert is started they could close automatically.
McCoy seemed to have as much trouble as the bridge during Journey to Babel
. What did he say to Kirk? Are you through shaking the ship around? So I would think if it's a powerful enough hit to rock the bridge all the ship rocks. I don't think having extra strength SIF to certain areas to be a very good idea. Correct me if I'm wrong, but if the whole ship is rocking every which way, and one section stays still, wouldn't the area just outside the SIF start bending, then maybe buckle and fail faster than if it shook with the whole ship?
And as for really cleaning up quarters that would be engineering and all ships personnel (except maybe the command staff and Medical) responsibility. Replacing fallen lights, restore power, getting those I beams that fall out someone's room would be a ship wide responsibility.