Does the 'day of many hats' theme become invisible to those who might be using a custom theme..Does the 'day of many hats' theme become invisible to those who might be using a custom theme..

Yes. The "day of many hats" is a theme, and custom themes override the AQ themes of the day. Someone may mention it on the AQ Board, but you're quite likely to miss the day if you have a non-rotating theme.