I am working on a 3D racing game using SDL and OpenGL. I thought it would be easier to use a 2D physics engine, since I really don't need the 3rd dimension. There will be no flying cars or jumps, they will just be stuck to the floor, so I would use 2D colliders and that things to simulate collisions in a plane but render the actual game from a 3D perspective. So the real question is: is it possible, is it a dumb idea, what else can I do?
It's not dumb, but realize that car/racing physics are very different from traditional game physics. You're not going to be able to just drop in Box2D.
Traction, handling, acceleration, braking, etc. are all going to need a good deal of special casing. It'll likely be a lot easier sticking to 2D collision handling, of course, but collision is hardly going to be the most complex thing you'll have to deal with.
It depends on what level of physics realism you are aiming for. If you're fine with a Mario-kart type of racing game, then you won't have much trouble with a 2D physics engine. But if you want a top-heavy vehicle to roll over when it goes around a sharp turn at high speed, then you will need 3D physics.