What effects would the moon have on our planet if it rotated more quickly on it's axis?

No noticeable physical effect. We’d see the entire lunar surface rather than just the side that currently faces Earth because it’s tidally locked.

Nothing, really. The centre of mass would still be in the same spot, and that's mainly what affects the earth.

However, the effect on the moon would be much different: Normally, the moon is tidally locked with earth, meaning that it always shows us the same face. This results in there not being a lot of stress on it. If it was rotating relative to us, tidal forces would be much greater, as these forces would tighten and loosen on points that would normally experience pretty static forces.

I can only speculate on the effects of this, but there would be a lot more mechanical erosion on the surface, and the body would be measurably hotter. I don't think it'd get hot enough to have a liquid core, as it radiates a lot of heat away; plus over time said rotation would most likely slow down to the point where it becomes tidaly locked again.