Does the flywheel engage the gear?

No, the flywheel is bolted to the back of the crankshaft. It is a large metal disk with teeth around the outer edge for the starter to turn the motor. There is a smooth surface on the flywheel face. when the clutch is engaged it is catching the flywheel so to speak, by pressing the two surfaces together tightly. This makes them one and so the engine is now turning the gears in the transmission, which ever you have selected at the shifter. A flywheel is only in a manual transmission.