Absolute velocity does not affect time, simply because there is no such thing as absolute velocity. Relative velocity, though, does. For instance, if you look at a clock on a GPS satellite, moving fast relative to you, you can see it run slower than the one on your wrist. Similarly, someone stationary on the surface of the sun would see us here on earth moving in slow motion.
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.
> For instance, if you look at a clock on a GPS satellite, moving fast relative to you, you can see it run slower than the one on your wrist.
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.
There's a really neat thing about this, the relationship balances out at a certain orbital height, before it flips over so there's an orbit where your chronologically in synch with the ground.
That's only true with respect to a given locus of points on the ground, not the full surface. I.E., the relative velocity of the SV isn't the same for all points that may be measuring.
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 velocity-time relationship is explained by Special Relativity (SR).
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.