Discarding your hand is a cost and the cost must be at least 1 card. So if you have zero cards in your hand, you cannot activate “Rope of Life”.
If your monster is controlled by your opponent and sent to your Graveyard, you can activate “Rope of Life”. You cannot activate “Rope of Life” when your opponent’s monster you control is sent to his/her Graveyard.
The increase in ATK is permanent as long as the monster remains face-up on the field.
You can use “Rope of Life” to Special Summon “Jinzo” and “Jinzo”’s ATK will increase by 800 points.