No, the key should not be hot. The key can be warm after extensive driving during summer. It's a possibility some parts of your engine bay is overheating or there's an electric discharge along the line but I am only speculating.
Let's work on the starter issue. Now, i've read what CuriousB mentioned about the AAA guy and the starter. He's right, the starter isn't engaged long enough to draw too many amps. I wonder if the AAA guy was trying to say alternator instead of starter, which is designed to draw and expel electrical current to and from the battery. That was going to be my third troubleshooting tip, replacement of the alternator, but before you do that see if your new battery has lost charge; that's one of a few clear indicators your alternator is defective.
So before you buy a new starter and the alternator, I would test the battery again. The Autozone and Advanced Auto guys have tester you can use or just ask them to run a diagnostic. A proper test includes testing with the headlights on, reving the engine and a couple of other things I can't recall of the top of my head. The final reading should indicate what device is going or gone bad.
I hope it's just the battery that's defective.