> its distance is also not limited by light travel distance
At cosmological scales, there is no single, correct definition of "distance." There is coordinate distance, light travel distance, angular diameter distance, luminosity distance, and other measures.[0] All these distances are the same at very small scales, but when you start looking at larger scales, General Relativity starts to matter and the geometry of the Universe is no longer Euclidean.
[1] https://en.wikipedia.org/wiki/Shape_of_the_universe
[2] see your own source
People quote different distances, depending on what they're using it for. Probably the most widely used distance measure, from Earth's perspective, is redshift. It's linear with distance nearby, and ask the other distances are relatively simple functions of it.
It is. See the source provided above. Technically the ΛCDM universe follows the FLRW metric, but if you look at its line element, you'll see it's usually just Minkowski in spherical coordinates times a scale factor (capturing the expansion) for spacelike coordinates:
ds^2 = -dt^2 + a(t)^2 dΣ^2.
Note that a scale factor that affects all spacial coordinates the same does not induce curvature. The generalised form of the FLRW metric includes a 1/(1-k) factor for the radial coordinate, but the curvature k for our actual universe is measured to be 0 within experimental precision, which, again, is a mystery. There's no good reason why a universe containing matter/energy should be globally flat (*cough* inflation). Factor out a(t) in a line integral and you'll get comoving distances over more or less euclidean space, i.e. the common format in cosmology literature. Things like luminosity distances are more interesting when you e.g. want to know the absolute magnitude of a far away object, but that doesn't really make sense for the CMB.
>Probably the most widely used distance measure, from Earth's perspective, is redshift.
Redshift is used to measure these distances sans any reasonable alternatives. But when you want to make e.g. a map, you can't use redshift (or proper distance for that matter), unless you want your map to be distorted.
You're confusing the flatness of a particular 3D hypersurface (of constant coordinate time) with flatness of 4D spacetime. The k parameter describes the curvature of that hypersurface, not the curvature of spacetime.
This is all to say that we don't live in a flat Minkowski spacetime. We live in a curved spacetime, which means that things like distance measures are much less straightforward. The curvature of 3D slices of spacetime is not the primary complication - the overall curvature of 4D spacetime (which involves the scale factor a(t)) is the primary complication.
That's not the case, because most objects we're measuring distances to are not at the same coordinate time as us. In fact, these objects are generally in the past.
If you're trying to measure the distance to an object that you've observed, that object is, in fact, along your past light cone. The only coordinate-invariant distance that one can actually define, the metric distance, is zero. That means you have to define an alternate type of distance. Some of those choices (like the comoving distance) don't depend on a(t), while some do (like the light travel distance).
> All of this is explained pretty well on wikipedia, but if you want you can also take a look at Hartle's or Carroll's textbooks on General Relativity.
I prefer Wald, which is much more mathematically rigorous than Carroll. I haven't read Hartle, so I can't comment on it.
Only after ~200 million years the first generation of stars appeared, which we cannot hope to see with a telescope. The first proto-galaxies (too faint) and galaxies took even longer, hence this billion years of darkness.