-To capture a VP point do you have to end your turn in it or can you just take it on the way to something else?
See 2.4 for controlling areas:
You do not have to end your turn in an area for changing control.
-Is sudden death victory possible for in the supply phases of both players or merely the supply phase of the second?
See "Extended Sequence of Play":
Check for Sudden Death is at the end of the game turn, not at the end of a player turn.