My friend is a long-distance runner and a pumper of iron. Running in particular takes a shitload of discipline and commitment, so if you're looking to push the limits of your physical capabilities, you could certainly get that done with running in nice surroundings. I would suggest biking, swimming and rock-climbing as well, but the consequences of possibly fainting or feeling dizzy in the middle of those activities is potentially fatal.
Any specific movements bring on your spells of dizziness/nausea? Would help narrow down what sports you could do - my friend is unable to bend her neck backwards without fainting, for one thing, so anything involving that is a no-go for her.
But getting a second opinion may be a good thing. Arnold-Chiari malformation is, as far as I know, usually a congenital thing and not really something you commonly develop from getting socked in the face, unless it was already in place at the time of trauma, in which case you may have noticed symptoms earlier, especially if you've habitually gotten smacked in the face before. Nausea and dizziness may result from something as boring as a common whiplash injury, so getting a second specialist to have a gander at your MRI scan couldn't hurt.