2004 Cadillac DeVille - New Alternator, Now Problems

Can you list the codes the car showed? Is the check engine light still on?

The no-crank when hot should not be difficult to diagnose the problem. I’d suspect the starter, especially if it has never been replaced, but replacing a starter on this car is expensive and a headache - it is in the vee under the intake manifold.

Rough idle sounds like failed Idle Air Control Valve (IAC). I had to look to see if this car had one, it does. If there is no check engine light, I’d suspect the IAC. They are cheap and easy to replace.