I find that behind the dash can get pretty warm when the heat is on, which is understandable considering there are numerous ducts in there. Enough that I put in a fan to keep the raspberry pi cooler. Never hit thermal throttling, but it was getting up there. Never noticed the front face getting warm though.
The stock head-unit (assuming they are all somewhat like the soundmark) has a tiny fan to cool the MOSFETS.
But, as you indicated, the lights are intermittently working, I’m going to go out on a limb and say there are other issues. I would definitely try and figure that out sooner than later, because if something i getting real hot.. that could end badly
Yes, the heater cores are directly behind the radio in the dash. If you leave the climate control on auto, it should always select enough airflow vs coolant flow to keep the cores from getting too hot.
I did have one factory radio that had problems with the button lights. I can't remember if it was the 2004 LS or the 2001 Grand Marquis. Either way, the problem was bad soldering on the button board and was easy to fix. It was not related to temperature or over heating.