It is not uncommon for the cut-off valve under the lavatory to get stiff with age. I have encountered cut-off valves that needed to be turned with pliers.
As to the hot water starting to run in the middle of the night, that is a new one to me, but possibly the valve had been on the verge of failing and then suddenly did so. If this is the type with a "rubber" washer seal, then a new seal may fix it. If it is the type with a "cartridge", then a new cartridge may fix it.
If you cannot get the cut-off valve to work with pliers (used carefully), then you will have to replace the cut-off valve.
If you have the water to the entire house shut off, then it might be a good idea to turn off the water heater. Then you may be able to turn off the supply valve to the water heater and then turn on the water to the house and have cold water supply at the taps. Having running cold water will allow the toilets to fill and you will have cold water at the kitchen faucet, etc. Then you can plan a fix for the hot water valve in the lavatory.
EDIT I don't see the left (hot) handle skewed at an angle. The position of the lever is a little different which it will be in general. Pop off the little cap unscrew the screw holding the handle down and remove it. Then remove the covering to expose the valve. Then remove the guts of the valve and I think it will be revealed as a cartridge. Take it to a big box or a plumbing supply and get a new cartridge.
It could be that the handle is poorly placed so that it does not close all the way. Remove the handle and reposition it so that when closed it is not in contact with the back splash guard of the counter top. That may be all that is needed.