will overheating damage engine

In general over heating an engine is a very bad thing! If your Malibu is the 3.1 V6 they tend to blow the head gaskets and intake gaskets after an over heat which causes loss of compression or worse coolant to get into the cylinders and lock the engine preventing it from rotating. I would recommend a compression test to see if you did damage the integrity of the block and heads. If the compression is good you will have to look else where to find your non-start problems.

No offense, but you should bite the bullet and call the man. Based on the reams of information you provided not much can be diagnosed, and I'm guessing that you don't work on cars much.

A compression check may, or may not, tell you something.

It might also be a head gasket failure because of overheating. Try to check the head gasket with your mechanic. He can replace the thermostat and check inside the radiator for any corrosion buildup or blockage.