When I turn the key once (without starting the engine), these (attached) lights show. When I start the engine, they all go off. No problems with car, but should the EPC, battery and engine light show like this or is it a possible fault warning? Thanks.
Yes. The reason that the light come on is to give you proof that the lights are working. If you are truly conscientious you look them over with great care to ensure the lights are all showing :-)
More seriously some of the lights are actually telling you something, for example the ABS and airbag lights are set on when the controller starts its self-check and then are turned off if the self-check completes successfully. If the light were to stay on it would let you know that the system required service.
Others are just logical. Low alternator output is one of the conditions that illuminates an annunciator. When the engine isn't running the alternator output is zero and the lights comes on.