Why? They're the only one's not nailed to the pole?
Why? Couldn't say why, can only say it is not done. At least not on any local, state, or national level that I know of.
[Citation needed] I'm pretty sure that anywhere a city's flag is flown alongside the state or U.S. flags, it is usually lowered when the others are.
This flag is under the direct control of the state legislature and has apparently always been treated as an official flag. The decision of whether or not to lower it is fully in the hands of the state government, and so the state government should be held accountable for the decision not to do so.
And the SC legislature doesn't get to hide behind this "well, it takes a 2/3 vote of the general assembly" nonsense either. Is there anything physically prohibiting the general assembly from getting off their asses and taking a vote to take it down?