Jonathan Goodish wrote:
Logic tells me that heat is created by electrical resistance, and I'm
guessing that there shouldn't be enough resistance in the master relay
to cause it to get too hot to touch. Is this a failure mode for these
relays?

Relays can fail such that they get hot. A relay that hot is probably a bad
relay. It needs to be fixed or replaced.

Could this be contributing in any way to my electrical noise
issues?

Possibly.