The order of play is
1. Move (mandatory)
2. Activate a system (optional)
1. In a single turn, can a captain activate a system that the first mate just readied?
2. For mines, activation means dropping the mine. The captain can then trigger a mine "at any time, except while surfaced". Can a different system be activated in the same turn that a mine as triggered?
As explained in this thread, my interpretation for this one is that blowing up a mine is NOT a system activation, and can be done at any time.
3. Can an enemy captain activate a system in the middle of the other team's turn?
Not in the turn-by-turn mode. But definitely yes in simultaneous mode.
4. Can an enemy captain trigger a mine in the middle of the other team's turn?
As for 2, this is my interpretation, but for me the answer is yes.
I will however come back with a more official answer soon for 2 and 4.