As the piston moves down the cylinder, the intake valve is opening. What governs piston velocity is the stroke, rod length, RPM, and piston pin off-set. You must be wondering why I'm talking about piston velocity during the first stroke. Volumetric efficiency is a measure of the effectiveness of an engine's intake system and there are about 200 miles of air above the engine just waiting to fill the cylinder with 14.7 psi at sea level. The intake valve is almost closed as the piston reaches BDC, but it does not close completely until after BDC, when the piston is on its way back up the cylinder.
Power is produced while the gases in the cylinder expand and cool.
Otherwise, the piston would have to push ALL the exhaust out. The exhaust valve is opening at a fairly rapid rate, the piston is going up, and if the exhaust valve is not open a lot by the time the piston reaches maximum velocity, there will be resistance in the cylinder caused by excessive exhaust gas pressure. This produces conditions which are referred to as pumping losses. At TDC at the end of the fourth stroke, both the intake and exhaust valves are open just a little. During the overlap period you will often find that both valves will be open an equal amount.