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"Confused yet? "

Well, that's the way I understood it. But then it seems like the simple train station math ought to work.

If you have the following:

E - Earth after 13 billion years.
B - Location of the Big Bang
X - Location of galaxy at 1 billion years after bang
D - Location of distant galaxy now.
S - The speed of the galaxies.
C - Speed of light
E<-----------B->X-------->D
(BE)(S) = 13 billion (according to scientists)
So (BE)(S) = (XE)(C) or the destance of the earth from the big bang time the speed the earth has traveled must equal the distance from point X times the speed of light.
Therefore,
(BE)S = (XB+BE)(C)
(BE)S = (XB)C + (BE)C
(BE)S-(BE)C = (XB)C
XB = {BE(S-C)}/C
XB = BE (S-C)/C
If S=1/4C then XB = BE (1/4 - 1) = -3/4BE
Or if Speed of galaxies = 1/4 the speed of light then point x should be 3/4ths the distance of the earth from the big bang. Meaning the distant galaxy would have to be 3/4 the age of the universe and that is as far back as we could look.
The faster the galaxies go the further we can look back until we are going the speed of light and then we should be able to see the big bang itself. But the slower we go, we shouldn't be able to look back very far at all.
The distance from the Bigbang to the earth times the speed of the earth
I know this is back of the envelope math, but it just seems counterintutive to what the scientists tell us about being able to look back and see a young cosmos.

The distance from the Bigbang to the earth times the speed of the earth

Ah, but there is no "distance to the Big Bang" It was everywhere, in a certain sense. Sort of like the light being everywhere as reported in Genesis. The light (or energy) of the big bang filled the entire universe as it then existed. Took a while for the universe to "cool" enough that matter could even exist. At the energy density of the early universe, only energy could exist. Your calculation are for a Newtonian universe not one where special, not to mention general, relativity exists, such as the one we live in. Even when dealing with solid objects, velocities don't add or subtract vectorially as the speeds approach that of light. If your math were correct, and we were moving away from some object at say 1/2 c, we would measure the speed of the light from that object to be 1/2 c, but we don't, we measure it to be c. (It will be rather redshifted to a lower frequency however)

When dealing with the speed of planets and galaxies relative to each other, simple Newtonian math works just fine.

Also, I agree that they are using some fuzzy math here. Supposedly the relative speeds of the galaxies are MUCH lower than c. If the galaxy they are talking about is 12 billion light years away, and the universe is 13 billion years old, then at some point nearly every object in the universe was traveling at or near the speed of light.

Also, using their argument that the light from the mentioned galaxy has traveled 12 billion years to get here, the simple fact that you can see it at all suggest that the relative speed of the galaxy 12 billion years ago was well within the Netwtonian physics range. Otherwise, the Doppler shift would have made the frequency of the light coming from the galaxy so low that it simply could not be seen.

Using those two points of logic assume the earth and the galaxy are relatively the same distance from the center of the "big bang" (6 billion light years) and that their speed is relatively low. That means that they traveled 6 billion light years each within the span of 1 billion years and thus violated the theory of relativity which states that no object can achieve the speed of light except light itself.

E<-----------B

Here's my take: I'm sure you've heard the oft-repeated analogy between the two-dimensional surface of an expanding balloon and the expanding three-dimensional space we live in.

Imagine this two-dimensional surface when it's no larger than the surface of a proton, say (or even infinitesimally small, if you wish).

There's a Big Bang. The surface begins to expand rapidly.

Fast forward 13 billion years: Some critters have evolved on the surface of the now gigantic balloon, and they want to know "where" the Big Bang happened relative to their position on the balloon. Answer: The Big Bang happened everywhere, at every point in space, on that infinitesimally small surface. It makes no sense to talk about how far away you are from that point now.