Yes, it’s the Axis player’s choice to make, in the first instance. Allies can do a few things to try to stop or mitigate it: (i) liberate and hold Normandy until France’s turn; (ii) put UK units in Southern France; (iii) but UK or US units in French territories that Allies don’t want to turn Vichy.
Simply putting a plane in Southern France probably wouldn’t be sufficient to stop it since Italy could simply attack with an inf, and its 3 planes. If it clears the territory of units without taking it, Vichy France will still happen on France’s turn.
Whether and to what extent it benefits Axis depends, of course, on whats happening on the board. In addition to the pro-axis infantry, Axis gets a free hand on Italy’s turn to do things other than deal with the French. Its gets a defensive fleet in sz 93, which, when coupled with a destroyer in 95, can serve as a handy blocker (not to mention a deterrence to allied landings in Southern France). The Vichy arrangement also potentially denies Allies a landing spot in Africa, since all the territories that planes could reach from the UK would go Vichy (unless UK takes steps to prevent this).
No, the French ships by Madagascar and the English channel do not turn Vichy.
Really the best way to get your mind around the Vichy France thing is to play. Up for a game?