Normal behavior? Yes, and very handy at times, though I wish I could find a way to avoid shutting off and restarting the engine just to remove the fob.
After shutting the engine off, yes, the radio will stay on. Just press the Start/Stop button once more to finish the job.
Oh, and not sure I'd use it for a valet if you have a MT. With my luck, at least, I'd probably get a valet who's not overly accomplished with a stick shift and have to retrieve my car from the middle of a nearby intersection.