How much would you have to change the rotation rate of the earth for noticeable geological things (e.g., earthquakes) to start happening as the earth settles into a new oblate shape due to the change in centrifugal force? I suspect small changes suffice: centrifugal forces will easily overpower the rigidity of the planet. How would one change the earth's rotational speed? Maybe launching flywheels into space.
Original thought: make launching satellites into low earth orbit easier by enhancing the speed of the equator. Current equator speed 0.465 km/s. Low earth orbit 7.8 km/s. If we increased the equator speed to match Low earth orbit, the earth would break apart, though bad things would happen before that.
The Earth's orbital speed around the sun is 30 km/s, pretty fast. But outer planets travel much more slowly. Are there planets whose surfaces or cloud tops are sometimes traveling backwards with respect to their orbit around the sun? Dropping something into the sun would only require escaping the planet's gravity well, not cancelling the orbital velocity.
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