GSC Pokémon (in general) on RS games: Maybe (downloadable as prizes?), but completely unconfirmed. (Almost 100% certain you'll be able to rent GSC Pokémon [probably with predefined movesets] though.)

Your old GSC Pokémon on RS games: Very unlikely - Pokémon Box isn't compatible with GSC and can't transfer between the two. Even if it was possible using the GameCube, I see no reason they'd choose Pokémon Colosseum to do it rather than Pokémon Box, since Pokémon Box is all about trading / storing / etc. So no for that.