I just observe that a lot of cosmology seems to be riding on the theory that the red shift is caused by an expanding universe.

This seems wrong to be. There's at least two independent lines of evidence for the Big Bang theory besides redshifts -- isotope abundances (particularly for light elements) and the cosmic background radiation.

What if it light just loses energy as it travels, so that the frequency shifts lower?

We would have to abandon our belief in energy conservation. And we would then wonder why energy seems to be conserved exactly in every interaction we can see. Also we would wonder why we see spontaneous redshifts not spontaneous blue shifts. Every known micro-scale physical process in the universe is reversible [1], and by the CPT theorem, we expect this to be true always. A lot would have to be wrong with our notions of physics to have light "just lose energy."

That seems like a perfectly natural solution. How do we know it isn't true?

This solution requires light from distant galaxies to behave in ways totally different from every other physical process we know about -- including physical processes in distant galaxies. It seems unnatural to say "the redshift is explained by a totally new physical process, and this process violates a lot of natural laws that hold everywhere else."

[1] I should say, reversible assuming you also flip the charges and parities. That's irrelevant here, though, since photons are uncharged and don't have any special polarization.

The actual theory, equation and everything is [...]

That's not the actual theory. It's a tiny fraction of a theory. It's not even clear that it makes sense. (What exactly is d here? Total distance travelled by the photon since it first came into being, I guess. But what exactly does that mean? For instance, can there be interference between two photons with different values of d, and if so what happens?)

In your theory, photons travel slower than c, the exact speed depending on their "d" value. That's going to mess up pretty much everything in quantum electrodynamics, so what do you put in its place?

In your theory, photons pop out of existence when their velocity and frequency reach zero. Again, that violates local conservation of energy and CPT invariance and so forth; again, how are you modifying the fundamentals of conventional physics to deal with this?

At c/H, in both the old and new law, the frequency of the photon reaches 0.

But "at c/H" the photons never reach us because their source is receding from us at the speed of light. We never see zero-frequency photons. We do see substantially-lowered-frequency photons (once the light has had long enough to reach us despite the recession).

In the old law, it remains to encounter ever increasing space, getting farther from the things in front of it.

This doesn't appear to me to be anything like correct. It is getting closer to the things in front of it -- apart from ones that are receding from it faster than the speed of light, which are (and remain) far away from it. What's the problem here supposed to be?

Old = photons piling up in ever increasing space

I repeat: I do not think the conventional theory does say anything like "photons piling up in ever increasing space". Of course it's possible that my analysis is wrong; feel free to show me why.