"As a QC/QP member, I assume that I will be able to use the Admirals Lounge at JFK prior to JFK/LAX flight. How about at LAX? As I'm arriving in Terminal 4, will I be able to use the Admirals Club there or will I need to use the club in TBIT?"

You will be able to use either lounge, though why there is such an anti TBIT feeling I don't know. There's nothing wrong with the lounges there

"I assume my luggage will be through-checked to MEL from JFK?"

They can be checked through and probably will be, though AA are under no obligation to do so since you are on 2 separate tickets

Dave