The Wikipedia article on sprouts, seems to suggest that it does not, although I can't see how to prove it.
It does not, it just depends on how many dots you start with. Informally, you would just argue that for any game you can move the dots around, stretching the lines as necessary to maintain the relations of inside/outside, to any pattern you like and get a corresponding game. More formally you could set up a homeomorphism between the plane with dots in one position and the plane with dots in another position and argue that the game plays the same either way. This last approach suggests correctly that you could make correspondences between partially played games that do not entirely respect the inside/outside-think of the plane as a punctured sphere and you can close it up and puncture it anywhere else to get a game with the same topology but with different inside/outside relations.