Re: cold water abruptly shutting off
It is likely a problem with the cold side of the faucet valve. If the faucet is a relatively decent one, then rebuilding the valve will be cheaper and easier than replacement. If it's a seriously cheap faucet, then I'd change it out for one of a better quality. I personally like Delta and Pfister the best. Stay away from faucets that are all plastic, they don't last or function well for very long.
I suffer from CDO ... Its like OCD, but in alphabetical order, LIKE IT SHOULD BE!!!