A gas check is a piece of metal crimped to the bottom of the bullet. Gas checks are usually copper or aluminum. The gas checks keep the high Temps of the Burning powder away from the lead bullet base. If a bullet does not seal entirely against the bore the gases will escape up the side of the bullet and melt the lead,thus leading the barrel. The gas check helps seal the gases to keep them behind the bullet.
Hardcast is mostly a generic term. There is no definite definition but refers to lead that is alloyed with other metals to make it harder. Those are usually tin and others. In some situations hard lead is not needed for a cast bullet at target velocities. Swagged bullets are soft lead. Hard lead allows the bullet to not strip the rifling. This means there is no slippage between bullet and barrel rifling. When bullets are accelerated very fast the bullet slides down the barrel and does not rotate with a 1 to 1 relationship with the rifling.
There can be one to multiple lube grooves on lubed bullets. The position,width,and depth of lube grooves determine the center of gravity of a bullet. Where the cg is determines the bullets stability. A bullet in a long rifle bullet requires more lube to seal and lube the bore since the bullet is in the barrel for a longer time than a short pistol barrel.