I was trying today to remove a kitchen sink. I tried shutting off the valves from the hot and cold water lines under the sink. The cold one closed successfully. The hot one clicked as it turned clockwise. When testing the faucet, hot water continues to come out.
Does that mean the hot water valve has failed? Or is there a way to fix it easily?

I can shut off water to the whole house, but in that case, how do I deal with the hot water heater being full of hot water?

Does that mean the hot water valve has failed? Or is there a way to fix it easily?
I can shut off water to the whole house, but in that case, how do I deal with the hot water heater being full of hot water
You can shut the hot water off at the water heater and unscrew the bonnet packing nut .(see image} and back the stem and washer assembly out of the angle stop. You may now replace the washer. Before you reassemble the angle stop place a pan over the open valve and have someone turn on the hot water for a moment to flush out the supply and valve. Reassemble and test. Good luck. Tom

What happens when the valve is open and no water is coming out - any idea on how to fix this? I turned it clockwise to close the valve (never really closed) and when I tried to open it up again, VERY slow water is coming out of the valve. Do you think it's so clogged up that the only way is to remove the plastic stem and clean and maybe replace this? Challenge is, I have to drain the whole hot water in the house :-( in order to do this...

You do not have to drain the water heater. Just close the input valve to the water heater. You can then repair or replace the stop valve under the sink. If the valve is a compression fit like the one shown in Tom's post above it would be best to just replace.

To make your job easier, if you replace the valve, use the old 1/2" compression nut and ferrule, Good luck, Tom