The rotations are set before the cruise. For a 4 night cruise the rotation are 1223, usually. Although 1233 has been done also.
That's Animator's followed by Enchanted Garden followed by Royal Palace. (followed by Animator's, etc, etc, etc). One third of the passenger load will start their rotation in each one of the dining rooms.
So the rotations are AEER; ERRA; RAAE or could be AERR; ERAA; RAEE.
You can call
DCL and request a rotation, but it's just a request. You may or may not get it.
As I said, Pirates night has it's own menu. And is typically on night 3 on a 4 night cruise. The other 3 nights you'll have the Signature menu for each dining room (different menu in each one).
The link I provided has a huge list of the various menus available. I'd suggest looking them over and making a request based on that, if you want.