|MadSci Network: Astronomy|
You're right that the Earth is moving very fast -- even near-lightspeed -- relative to some places in universe. For example, because of the expansion of the universe, someone at a very distant star might see the Earth receding from them at near-lightspeed. They would therefore see our time go by very slowly: if they had good enough telescopes, they might see the Earth take thousands of years to go around the sun just once! As far as those observers are concerned, time on the Earth basically HAS stopped!
But how other people at other stars might perceive the Earth doesn't affect how WE perceive the Earth -- it's all relative! From our reference frame, the Earth goes around the sun in exactly one year. There aren't any relativistic effects from our perspective, because nothing is moving at near-lightspeed.
Also keep in mind that there's no such thing as a "stationary" reference frame where you can measure absolute speed -- all motion is relative.
If you want to know more about how the very same events can appear to take different lengths of time to happen (depending on your frame of reference), try some of these relativity links.
[Moderator's note: the speeds given in the question are not correct: the speed of an object on the equator due to the Earth's rotation is about 0.5 km/s, and the orbital speed of the earth around the sun is 30 km/s.]
Try the links in the MadSci Library for more information on Astronomy.