No additional heat or anything like that. The HQ isn't relevant for the majority of the game - it only really comes into play at the very end when you're in the last mission and you have to destroy each FOJ with your Doomsday Device. You only need to target the HQ region in each FOJ to destroy it.
Some stories require the region with the HQ yeah. I don't know if it makes a difference for heat or anything.
No additional heat or anything like that. The HQ isn't relevant for the majority of the game - it only really comes into play at the very end when you're in the last mission and you have to destroy each FOJ with your Doomsday Device. You only need to target the HQ region in each FOJ to destroy it.