Flat wings can and do fly just fine, they are just a bit less efficient. The teardrop shape and camber/cupping underneath just make the winds more efficient at slicing through the air without creating as much turbulence and drag.

I think this is actually a good explanation of the lift force, but lots of other factors come in to play for wing shape. Two other big factors are drag forces, which are dependent on the surface area, air density and the velocity of the craft, and so there's a complicated optimisation problem there, and turbulence, which depends a lot on the wing tilt, and the the shape of the wing.

That is, in fact, true. Planes can fly with symmetrical wings, including flat wings, and even with upside-down wings, as is easily demonstrated by observing stunt planes actually flying upside-down--in all cases, as long as they are appropriately tilted to redirect the air. Most purpose-built stunt planes even actually have symmetrically-curved wings, because the reduction in right-way-up aerodynamic performance is made up for by the increase in inverted performance.

Paper airplanes fly, and they have flat wings!

It turns out that flat wings work just fine, but the airfoil shape we see on airplanes is more efficient:

http://warp.povusers.org/grrr/airfoilmyth.html

I'm not too sure it won't work, but I'm pretty sure it won't be efficient. Or maybe your plane will just rotate until the wings are horizontal again.

Flat wings can fly. E.g. kites.