using basic math and logic, i have proved that there is changes in velocity that occur on the surface of the earth.

The example you offered requires considering other forces. As you mentioned, speed of earth through solar system >> speed of rotation. Scientists have observed minute differences in perceived mass (ie gravity) at places such as on top of mountains vs desth valley. Use your string example and consider what would happen while moving in a car. The length of string would have almost no overall effect on the overall notion of the object to an outside observer.