Europark can store your car keys. The service costs 50 NOK, and the office is open at all hours. Europarks office is found in the terminal building, at your left when you go through the main entrance.
It is unfortunately not possible to store luggage at Bergen Airport, Flesland. However, it is possible to check in until 24 hours before departure if you travel with Norwegian, SAS, Widerøe or KLM. Assuming the desk is open.