How can I prevent my furnace and A/C from slamming doors?
This is common in HVAC systems with a single central return. My system does this too. It's simply the nature of a central-return system. The only practical way around it is to have a balanced system with per-room returns sized the same as the supply registers on those rooms.

There are a couple of things you can do about it. One is to spend hundreds to thousands of dollars necessary to re-do your house's return ducting to put a dedicated return in every room. Another thing that might help is to replace your doors with heavier ones on  less-well-lubricated hinges. Or you could just keep the doors wide open.

As for the whole-house fan, remove it, seal up the hole, drywall the ceiling, and pile insulation over the top to the level of the rest of the insulation on your attic floor.