Time in time-dependent theories

I have this odd question. I'm not a physicist, just a computer scientist in training with a broad range of interests. So excuse me if I get something completely wrong.

There seems to be a bunch of theories in physics that basically say that some or all universal "constants" are not in fact constant, but change over time. Everything from Dirac's seemingly numerological equation about G to interpretations of recent observations. Basically these people claim that some or all of these "constants" are tied to the age of the universe.

Then there's special relativity and time dilation caused by high speed and gravity. As I understand it, special relativity has been experimentally proven to be accurate.

Is it just me or are these two ideas incompatible?

Consider, for example, a particle moving at c or near c that was set in motion when the universe was still young. Time dilation would say that less time has passed for that particle than for the rest of the universe. Therefore, the "age of the universe" would be different for that particle.

Or consider a black hole. Time dilation would again suggest that time slows down near it and so the "age of the universe" measured near a black hole would be different yet again.

But if the "age of the universe" is a variable in the equation for some constant, then that would mean that constant is different in different areas of space.

The whole concept of the "age of the universe" seems wrong if time is really relative.

The purely conceptual "bias drive" spacecraft propulsion system developed by NASA in 1996 operates by changing the gravitational constant G in front of and behind the drive. Allegedly this would cause a singularity to form in the gradient of the field -- inside the spacecraft.

Doesn't this mean that if G is based on the age of the universe and the age of the universe can be measured differently in different areas of space, then G would also be different in those areas since there is no absolute time and if the theory behind the bias drive is correct, wouldn't it also mean there'd be singularities popping up all over the place?

Or if you take it that some or all currently accepted constants are still dependent on the age of the universe and that they are the same everywhere right now, then wouldn't that mean there's an absolute frame of reference? Or at least a hidden variable -- a hidden time dimension -- that is not affected by time dilation.

As I said, I'm not a physicist and here I'm using a software engineering approach -- simply putting together bits I've found to try and get a working system. It's just that many things seem to break down if you make some or all constants dependent on the age of the universe.

Or should I simply ask: is there any credible observational evidence for the claim or any mainstream theory that claims that some constants are not constant after all?
 
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