Why the pushrods? From what I cann tell, the compressor already comes with valves in place. I wonder if I can get away with using those.
No, because of the way an engine works as opposed to a compressor.
An air compressor builds pressure in the tank. The cylinder is just a pump to move air into the tank. Compressor valves are merely one-way valves that keep the air moving the right way. At no point in the compressor's cycle is there enough pressure in the cylinder to enable combustion, and if there was, the ignition event would open the exhaust valve. Disconnect the tank and the compressor isn't a compressor anymore, it just moves air from one side of the room to the other, like a very noisy fan.
4-cycle engine works this way:
Intake: piston moves down with intake valve open, cylinder fills. OK, this part works with compressor parts. Not optimally, but it works.
Compression: Piston moves up, both valves closed. Not how a compressor works. As a compressor's piston moves up, the intake valve closes and the exhaust valve opens, letting air into the tank. There is no cam to do this, they're just spring-loaded reed valves. Without compression in the cylinder you don't get...
Ignition: Spark ignites air/fuel mixture, both valves remain closed. See paragraph above. Also, you need to put a spark plug in there, as well as a way to fire it. Will the reed valves take the heat of combustion? No, they will not.
Exhaust: Valve opens, piston moves up to expel the gas. A compressor does this.
So of the 4 cycles, a compressor can do two of them.