Yes you can make it infinitely large. Or you can turn off depth writes, draw the skybox, then turn on depth writes and resume, which will ensure the skybox appears behind everything. Then the actual size of the skybox is irrelevant.
I'm not sure what you mean by making it transparent. Surely you don't want to see "through" the blackness of space to something behind it?
As for the textures, for star textures you can probably get away with using the same one for all six cube faces, unless you want there to appear a nebula or galaxy or something like that in the distance.