We do know the age of the universe, to a reasonable certainty. The discussion above doesn't argue against our current estimate of the age of the universe -- it addresses other issues.
A star moving in the opposite direction of us that emitted light 15 billion years ago that we are just now seeing has moved a lot further away in that 15 billion years it took that initial light to reach us.
It doesn't work like that. Yes, a photon travels at 299792458 m/s, but photons emitted 13 billion years ago (a redshift of about z=8.6) travel much further than (1310^9 years 3.1510^7 seconds/year 299792458 m/s) due to the metric expansion of space.
The correct calculation involves an integral, an example of which you can see here:
Given that we already know that originating at point A and traveling to point B, such that the distance from A to B at the time you arrive is X, does not involve traveling a distance of X... I have trouble seeing your point.
> If we receive light that traveled a distance of X light years, then the event is occurring X years in the past.
Not necessarily. By the time the light arrives here, its point of departure (and parts of its path through space) may have receded because of cosmological expansion, so its total travel distance, and its velocity, are no longer synchronized as they would be in Newtonian physics.
Your remark would have been true for the static universe that Einstein believed to exist when he proposed the first version of the cosmological constant (before 1929).
