What gets damaged when an E36 runs out of gas, and both secondary air filter valves melt?

Secondary air filter valves??

how hard is it to change out the secondary air filter valve?
what a bright idea, wonder what made it go bad?

It isn't difficult to change, but that seldom has any affect on the car running properly. The only time when it would make a difference is if the vacuum operator line is open. That would be a minor intake leak and could cause a very slighty rough idle.

A plugged fuel filter or a dying pump can be a consequence of running the car out of fuel.

You most likely clogged a fuel filter by sucking up debris that was in the very bottom of the tank. That's why you shouldn't let a tank go dry, aside from the obvious problem of having to walk to get some more fuel.

When you let it run out of gas there's a high probability of fuel pump failure on an older car.

about the debris being sucked form the bottom of the tank. The fuel is ALWAYS sucked from the bottom of the tank, THIS IS A MYTH. You do have a higher probability of burning up the pump by running the tank dry though because the fuel in the tank cools the pump. when no fuel is in the tank the pump has a greater chance of overheating and failing. As for the vehicle not staying running. doe sit just not idle or will it die when you hit the accelerator. If it won't idle you may have a idle air control valve goin bad. if it dies when you hit the accelerator it may be the pump.