I know that when a Fleet Action instance is completed that it normally take around 5 mins for it to reset. However, how long does it take the quest that goes along with the fleet action to reset. In other words, when can I redo the fleet action quest so that I can get some extra experience?
I think it is either a time based reset or a "leave the sector block" style reset.
As far as I am aware, you can only get the mission once. The primary reason to repeat fleet actions (besides that they're fun) would be for the loot you get as part of having a high score.
For example, I managed to come in at number 1 on Starbase 24 this morning (I am not really sure how, but it was nice) and received a very nice blue Impulse Engine which I promptly had my ship's chief engineer install.