If the cap is lifting before the design temp then obviously pressure is entering the coolant system from somewhere.
The fact that coolant flows into the overflow tank continuously while running and faster when you rev the motor seems like a cylinder to water jacket leak of some sort (head gasket, cracked head, or cracked cylinder). Does the flow into the overflow react nearly immediately with the increase in revs?
While the compression of the engine does not change when the engine is revved, the motor is essentially a positive displacement pump. The faster it revs, the more air it moves and thus pushes exhaust gas into the coolant at a faster rate.
It sounds like you have some motor work in your future :-(