Cold Water Line frozen, should I leave cold water faucet open?
It makes absolutely no difference whatsoever to the freezing condition if you leave the faucet open or not.

If a faucet is left open BEFORE it freezes, it can prevent freezing because running water does not freeze.

The only way to thaw it is to find the area that is frozen and heat it with a warming strip or a hair dryer or a propane torch or whatever. Try not to start a fire. If you use a hair dryer it can take HOURS to thaw a water line, even if it is blowing at high right at the frozen spot. Also, a hair dryer will only work if the line is frozen in one specific spot. If, say, 3 feet or 4 feet of piping are frozen, there is no way a hair dryer will do anything (at least in a day or two).