Maybe the tank somehow drains, but then doesn't fill when the bath or the shower is running, but there IS water in the bowl.

Scenario: Go in Bathroom, Turn on Shower. Use toilet. Go to Flush toilet, but NOTHING happens. Toilet lever IS attached, but the tank is empty and won't seem to fill.

I've seen questions where the OPPOSITE happens, but this isn't the case. The toilet fills and flushes fine as long as the SHOWER/Bath Faucet is NOT running.

Oh, and BTW, we've got that constant "whine" in the house that probably has something to do with air in the line as well. Perhaps the issues are related.