Actions on Ships

Anything. The player gets an action and the ship/fleet gets a seperate action.

Players can go on overwatch while a fleet moves, or refit their MS while a fleet moves, or even spar each other while on a moving fleet.

This is already in the Appendixes thread, FYI. Appendixes - Misc.

QCan you move onto a moving fleet (a non-action) and then perform the action as if you were already on the ship?
AYes.

QWhat happens if you're ambushing someone in an action, apart from the ship's movement, and you get damaged, and the patrol catches you?
AAttacking somebody that the ship/fleet is not attacking moves you off of the ship/fleet. Technically, you cannot attack somebody who is not at the same position as you. If you're on a ship, your position is the ship, and your enemy is not at that position, so you have to move off the ship (non action) and then attack.

QDo you automatically get stopped but the fleet can fight past? What if you're on overwatch? Do you get to ambush the patrol?
ANone of these questions make sense, because you cannot attack something that is not in the same position as you.

QCan you paradrop onto a patrol? If so, what happens if they rolled to stop you?
AYes you can. However, as per the paradrop rules - "If a Paradropping ship is attacked only the movement part of the action is made." So you make the move, patrols are rolled. If you get hit then you fight a patrol battle (patrols ambush you, if they win your movement is cancelled as well.) If they miss you, you make the move and then ambush the target patrol.