Actually, the general relativity effects of weaker gravitational field dominate the special relativity effects of velocity[1]. So the GPS satellite clock actually runs faster, not slower.
[1] http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps....
In practice the GR effect is compensated by the satellite at manufacturing, the SR effect is treated in the receiver - for just that reason.
The Sun example is a bit more complicated, because there's the velocity-time effect due to SR, but also the fact that you're deep in a big gravity well, which has effects on your measured time due to General Relativity (GR).
SR says that "moving clocks run slow", but deep in the Sun's gravity well, we should also be running slow relative to the Earth. Not sure what the relative size of the two effects is.
EDIT> I assume the SR effect is larger than the GR effect, simply because SR was obvious before GR.
However, someone moving on the satellite would see the clock on the satellite move normally, and us on earth moving in slow motion, because they are on the same frame of reference of the satellite.
So no, time does not operate differently on different parts of the universe per se. It all depends on how it's moving relative to the observer.