Does the rotation of the earth affect gravity the of objects on its surface?

The rotation doesn't affect gravity per se, but it does cause an extremely minor apparent centrifugal force that acts in opposition to gravity, thus reducing its effects a bit.

If the rotational speed were to change abruptly, the vertical effects would be trivial compared to the horizontal effects, which would basically be an earthquake with a Richter magnitude of 100 or more!

The rotation causes a slight bulging at the equator, and a slight flattening at the poles, so it does affect the surface gravity. You weigh slightly less at the equator, and more at the poles by the inverse R^2 law.

No - the gravitational field, strictly speaking incorporates teh rotational effect.  If you use the Newtonian model fro gravity, then the force of gravity is only dependent on the mass of the two bodies (Earth and you for instance) and their separation of their centers of mass.  Rotation off the earth does not alter the force of gravitational attraction but it does reduce the net force on a body on the surface of the earth.  There is a centripital force that points opposite that of gravity so the two forces net out to something less than Newtonian gravity would predict.   However, at the equator you only get 0.33 m/s^2 acceleration opposite that of gravity (9.8 m/s^2) so it is really negligible.

If you go to the full blown relativistic theory of gravity, then the rotation is completely coupled into the gravitational field and you can't really ask the question the way you did.