I'm trying to do this project for a class, and I really need to know that if you bend any given lens if it will increase its power.

Look up the lensmaker's equation (Wikipedia will provide) or the thin lens formula if you can pretend your lenses are thinner. Work out what happens to the radii of curvature as you move the lens edges while keeping the centres still, and see what the maths says.

The general answer is beyond my in-my-head maths at the moment, but an obvious special case is a plano-convex lens distorted so that the curved side becomes plane. The once-plane side must then have the same radius of curvature as the once-convex side had. That means that the power must be the same in that case. I don't know about other deformations or other lenses, so I'd suggest doing the maths.

Does that make sense?