I think what is leaking is the drain. Does it look like the object sticking out to the side in the below picture? If so, this is used to drain water from the house side of the supply when the valve is off.
It sounds like you over tightened it in the first place which started the leaking. If you've broken only the cap, you might be able to find a replacement. If the stem and threads are damaged, long-term you will need to replace the entire valve. As a short-term fix, you just need to block it some how. Marine epoxy is handy for emergency plumbing repairs.
The heater is leaking.
Therefore, it does not hold water.
Water heater electric heating elements die a short, nasty death if they are powered up in air, rather than in water. Gas water heaters will burn out the bottom if the flame comes on with no water in the heater.
You should shut off the electricity or gas BEFORE you shut off the water, or very, very shortly thereafter, both to reduce damage to the heater (which may not matter much if it's leaking in a way that means it needs to be entirely replaced) and also to reduce risk of fire when heat is applied to a water heater not full of water.
If you are in a place where your water heater might freeze you should STILL turn off the heat source (electric or gas), and then drain all the water from the tank (and consider what a bad idea it is to locate your water heater where it can freeze in the first place, but that's an opinion...)
For most repairs and any replacement you'll want to drain any remaining water (amount depending on the location of the leak) anyway, but that's easier if you wait until it's cooled down some.
Best Answer
Hot and cold lines can communicate through single lever delivery faucets. But the shutoff valves will stop this. Turn off the hot supply at the shutoff at each single handle control. Usually it is the kitchen faucet doing this, but it can be any where.