When Earl falls off the cliff, his hand is severed at the wrist. A second later, in mid fall, Earl's hand is fully visible.
The cop is dead but watch closely when Earl collapses for the first time. For a few seconds you can see the cop still breathing despite the fact that he is dead.
After Earl kills him, the policeman falls through the broken window. Earl could have easily used a shard of glass from the broken window to either cut off the officer's hand or cut him open and retrieve the key to the handcuffs.