I'm pretty sure KH is designed so that anything's possible in terms of gameplay. However, looking at Re: Coded, it would be EXTREMELY difficult to beat it at level one. Like, you'd have to have insanely mad DS skills. So, if you have those skills, I'm sure you can beat it at level 1.
It definitely is possible, I've tried it before but never completed it. Olympus Coliseum is almost impossible though due to a bunch of different variables (never truly being able to take 0 damage), but once you finish that with some luck, yes, you can.