The spy can have his cloaking ability. Would work for getting a sneaky shot or brake-in-front maneuver. Could work for screwing someone out of 1st-3rd place.
Pyro can also have a Lamborghini Diablo as his upgraded car, and a FIRE-truck as his first.
Medic can have an ambulance, of course menacing with spikes, and a Formula-1 Syringe as his upgrade.
Demoman, can have one of those metal hamster ball kind of things, decorated to look like a bomb, or a nuclear symbol painted on it.
Scout can have a modified golf cart as his upgrade.
Engineers, I think should be able to plant some hindering devices (for team races), like a device that can temporarily screw around with the turning or acceleration speeds of the other players, or even reverse the controls.
Spy, see above.
Medic; drop med packs or ded-packs (careful, both are indiscriminated in which team they affect; but the team he drops for can tell the meds from deds apart that were dropped by him, but not the opponents. Perfect for traps.).
Of course, to not abuse everything, the ammo for each should be really limited, and ammo packs lie around (also an engineer ability to produce; of course, like medic, there are good ones, and bad ones that can be deposited.)
That's all I can come up with for now.
As for a map idea, I know someone might end up replicating some Mario Kart levels (complete with 2D layout with invisible walls to back it up, like the classic SNES version). But to be more serious, I think some of the original levels that came with the game can be adapted for this mode, provided proper tweaking is made.