if you are having a power loss on one system while you put on a power load for another system then your electrical system can't handle the load. ( obvious part.) this is due to a couple different things. Battery is shot, may have worked one day but if it had a bad cell you wouldn't know until it all of a sudden gives up. Bad charging system, if the alternator can't put out enough juice this can happen. If the grounds are not making good contact you will have a resistive ground point, which means either all the current is flowing through that one point and the resistive nature of it is causing a low current flow, which can make the car act this way; or since electricity like the path of least resistance, one bad ground can couse the current to go to the other grounds, which if they are too small this again limits current flow.

since this is involving so many different systems if it were a cruched wire bundle you should be able to see it easily. if the grounds for these were shorted ( connected to ground in a different spot) it wouldn't make a difference, ground a wire at the end or 2 feet up the wire, as long as nothing is in between it dousn't matter. if the hot side of the circuit were shorting then you'd blow fuses from power going straight to ground. If it's not what I mentioned above the only thing I can think of in addition is if something punctured the wires bundle and crossed the wires, which when the puncturing item is withdrawn you'd have a small hole and several crossed wires.

If you can get it to autozone have them run a charging system diagnostic to rule that out.