2: When I'm drawing a skybox, should I have it drawn relative to the player or to the map?

Nobody answered this one yet

Skybox should move with the player. But only translation; rotation of the skybox must be fixed with the map.

The reason why you want to move skybox with the player is that this way it will appear to be really far away and you don't need an extremely huge skybox model. If the model wasn't incredibly huge and if it didn't move with the player, it would appear to be very close, which would break the immersion.