Trying to count the total number of galaxies in the universe is a difficult task, made harder by the part where no one wants to spend an infinite amount of time counting galaxies. Instead, what we usually do is count the number of galaxies in a very small area of the sky. Usually what happens is that we point a telescope at a very empty, dark patch of sky, and wait for a while. We’ve done this a few times with Hubble, creating what we now call the Deep Fields. We now have the Hubble Deep Field, the Hubble Ultra Deep Field, and the Hubble eXtreme Deep Field. (Once more, astronomers prove themselves eminently practical namers.) Once we have a really deep image, we can then assume every other patch of the sky is roughly going to look the same (as far as we can tell, a valid assumption). We can then multiply the number of galaxies in that one piece of sky by the fraction of sky we looked at, and get a very rough estimate of the total number of galaxies. Hey presto: several hundred billion galaxies!
But you’re absolutely on the money with your suggestion that we can only see the galaxies whose light has reached us. This fundamentally limits our observing, so that the galaxies which are most distant from us are also very far removed from our current time.
And here we begin a giant puzzle, because the most distant galaxies, which we see the farthest in the past, are very different from the galaxies we see in our local, nearby universe, which we see much closer to our current time. This means, again as you suggest, that the ancient galaxies we can spot in images such as the Hubble Deep Fields, must have evolved and changed between the time when the light we observe left them, and now. Galaxies which were once independent will have merged in the billions of years which have passed while that light was making its long way to us. Some galaxies will have used up or lost the gas they need to create new stars – one of the few ways a galaxy can “die out”, though its existing stars will disagree with you on how dead they are. Some galaxies will have uneventful evolutions, though they will still evolve. At a base level, galaxies will be creating more stars over time and adding to their own mass, though the number of new stars they make each year will drop over time.
Untangling the complex line which can connect a nearby galaxy to the sort of galaxy it might have been, billions of years ago, is a whole subfield of astronomy, under the moniker of galaxy evolution.
It’s important to keep in mind that it’s not quite as simple as seeing the same things repeated over and over again. The galaxies we see much earlier in their lives than our own are truly, physically, very far away, which is why we see them so far removed in time. They will be evolving over time in their own physical space, but they should evolve into something that looks like the galaxies near us. Distant galaxies seem to be the same everywhere we look, so we shouldn’t be looking at a special group of distant galaxies that would evolve in a unique way. They’re not the same galaxies as the ones that built our own galaxy, but they should be pretty similar. It’s up to us to learn what the pathway between ancient and current day must have been.
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