Not sure about the valet tag, that would be a question for the agent.
Cockpit space on the Md80/737 is very tight, it can be done but one of the pilots won't be able to slide their seat back. I personally would find a way - set the bag on the floor under the jumpseat - but I'm probably in the minority. As far as the forward closet goes, the Md80/737 forward closet is used for the cockpit crew bag storage and is usually full already by the time all the overhead bins are full.
I too have always been able to stow my bag in the cockpit of USAir Airbuses but then again, it's a roomy cockpit. Same with us on the 75/76/77 - no prob at all stowing your bag up there, I've done it a bunch for jumpseaters. It just may be too tight on the 80/737.