rotation - If Earth didn't rotate, would we feel heavier?

The centrifugal force at the equator is only 1/289 of normal gravity, or 0.35%. That's too small to be perceived.

http://books.google.com/?id=mE4GAQAAIAAJ&pg=PA358&lpg=PA358

It gradually diminishes as you move away from the equator, and becomes zero at the poles.

In any case, if Earth stopped rotating, the added weight at the equator would be the least of your worries. There would be major perturbations in terms of weather, etc, that would amount to global catastrophe.

Likewise, if Earth's rotation increased, would we feel lighter as
centrifugal force lifts us from the ground?

In theory, yes. But the rotation increase would have to be huge. Again, in that case, there would be other, much bigger effects, that would be much more important than your loss of weight. E.g. the whole planet would bulge at the equator and become more flat at the poles, with incalculable effects on plate tectonics, weather, etc.