Hi Mike,
Yes, with rare exceptions, all cabin doors, plus drawers, cabinets and lockers were fitted with keys. Generally they are what Americans call "Skeleton" keys: keys with a machine-turned shank, an oval head and a paddle at the business end fitted for wards and two or three tumblers. The pattern is still widespread in the UK today, although there the term skeleton key means a key modified in such a way as to force open a large number of different locks and so is usually used by
criminals in lock picking.
In First Class, passengers were issued keys to the entrance doors. In addition, there were locks on the doors between the rooms in a suite, however, each side of those intercommunicating doors was fitted with a double bolt that required both sides of the door to be unlatched in order for the door to open. This was done because frequently, passengers would only engage single rooms in a suite, and the bolts prevented inadvertent entry.
In Third Class, the cabins were likewise fitted with door locks, however, here it is questionable if keys were handed out automatically to the passengers. It may have been on request since it was not uncommon for 4 or 6 strangers to be staying in a cabin. That's a lot of keys to keep track of and the mindset of the time would probably question the need.
Cabins were keyed so that they could be opened by the cabin key itself, and a "local master," i.e., a key kept by the bedroom steward of a particular section so that he could get into cabins for cleaning, etc., but yet did not give him access to all cabins on board for security reasons.
The main key locker (for spares and duplicates) was under the main first class staircase on E deck.
I think the LA exhibit has a flat stock stamped key, typical of American practice. I suspect that it was being carried by a passenger onboard and is not part of the ship's outfit.
So far as I know, the only flat-stamped keys were used in connection with the ship's safes and deposit boxes, which are of a characteristic profile.
Bill Sauder