There's no roster moves during the draft (other than your drafted players arriving on your roster) - players you cut will leave your team at the next free agency stage, and players you offer contracts to will evaluate them at the next free agency stage.
When you queue a player to be cut, he is automatically removed from your active roster and you can't put him back on your active roster unless you cancel the cut. If you're on the depth chart page, you'll notice that they don't have the '+' button to activate them. They will also be listed in the "pending contracts" area (My Team->Pending Contracts)