At the moment, I named the headquarters as Tython.
For the last thing, I'd like to hash out what the Grand Master and Master of the Order actually do - that way someone doesn't go crazy or has no idea what to do.
Looking at canon, the Grand Master served as a spokesperson and representative rather than an executive. He/she also served as an adviser to the Supreme Chancellor. I can't find many instances of where the Grand Master makes decisions without the input of the Council, and the Council also had the ability to remove the Grand Master from power if he/she went overboard. Basically, he served as the voice of the order and held great respect, but he wasn't a dictator that could act without support.
There isn't much I can find on the Master of the Order, but I imagine that he would be one of the people that can officially call
the Gathering and decide on the topic of the meeting. More than that might go against the point of having all members of the Council standing equal.