If the gamecube is outputting progressive component (assuming a US or Japanese cube), then its 60fps progressive.
If its interlaced output, then it'll just output what the TV expects to receive - i.e. 60 fields per second, two consecutive fields make up a frame. However, if its a fast moving game, there is no guarantee that the second field will match the first, which is why you can get some combing on TVs. Its not the TVs fault, its because the console is running at twice the TVs full frame refresh rate.
It doesn't do any pulldown itself, thats up to the projector to deal with if its an interlaced source. It should cope fine in video mode, which assumes no pulldown required, just matching pairs of frames.