Half my house losing power, HVAC the key factor
If you're in the US, my guess is that you're losing one leg of your incoming power somehow.  Power comes into your home as split phase 240 volts - there are two 120V "legs" that are 180 degrees out of phase.  Typically one leg goes to each side of the breaker panel.  I bet if you trace the circuits that are down, they all go to the same half of the panel. When the air conditioning, which uses both legs, turns on, it bridges the gap and backfeeds power to the other half of your house.

Now, how you're losing a leg of your power is another question entirely.  This is likely a utility problem and you should call them ASAP. The utility will check things out and if it is on their side of things then they will fix it for free. If they determine that the problem is in your panel, they will let you know and then you need to call an electrician for a repair. This is an unsafe condition and can cause all sorts of problems, so get it checked out ASAP.