When flags are flown at half mast, is the US flag still displayed higher than a state flag?

As many are today, flags are at half mast. But not all. So, actually this is a threefold question. Is there an “official pronouncement” to display flags at half mast? Can you fly the flag at half mast whenever you personally feel it is justified? And the original question, if flags are flow at half mast, is the US flag still supposed to be higher than the state or other flags? Except for Texas, of course