I did disregard gravity since the density of the universe is pretty low on a large scale, so maybe there are some corrections to be made to take the (small) curvature of the universe due to gravity into account, but I think most of my explanation is still valid. If you can just imagine a universe without (much) gravity, I don't think there's anything wrong with the models I described. I will just give them again with a bit more detail:
The first reference system I proposed is an extremely simple one (but with complicated consequences) constructed along the lines of special relativity:
1. Pick some "stationary" point in our vicinity as the origin (of course not the earth or the sun, since they are revolving around the center of our galaxy, but some point in our local cluster of galaxies should do fine).
2. Using a set of standard "measuring rods" (just like Einstein used in his popular description of relativity), which are placed stationary to the origin of the reference frame, construct three perpendicular axes to assign space coordinates.
3. Place a clock at the origin which sends out a signal every second. Clocks are also placed at various other places in the universe and synchronised by listening to the origin clock and applying a correction of distance divided by light speed. A clock located one light second away, for example, would add one second to the received time to indicate current time in our reference frame. All these clocks are stationary in our reference frame, they are not moving with the expansion of the universe.
In this reference system, distant galaxies are flying away from us at high speed and are therefore subject to lorentz contraction and time dilation. They will appear shrunk in the direction of motion, and will age more slowly. They are also closer together because the space between them is lorentz-contracted as well. Obviously, an alien civilisation in those galaxies would consider themselves to be the center, and would say we are aging more slowly than them, but that's just the classic twin paradox and no contradiction. We are free to use our own reference system.
Anyway, using this reference frame, objects at large distances are younger because they have been aging more slowly ever since the big bang. At some distance away from us, objects are traveling at speeds close to c and they are so young that the big bang has only just happened for them. A little bit further, at c times the age of the universe, there's a singularity where the big bang is happening now.
Of course I realise that this is a strange way of looking at the universe, but it obeys the principles of special relativity (disregarding gravity) and therefore has the property that nothing exceeds the speed of light relative to the origin.
The cosmological model, which is a lot more practical to use, is constructed in a different way. Clocks at any point in the universe are set so they indicate the amount of time passed since the big bang as experienced by a local observer that is moving with the expansion of the universe, and distances are measured so that local light speed, relative to a local observer moving with the expansion (i.e. relative to "expanding space"), is c. Now the universe is the same age everywhere, looks the same everywhere (no more lorentz contraction from the expansion), but we gave up the property of nothing being able to go faster than light globally. Local light speed is c relative to local observers who are moving away from us, or in other words, relative to "expanding space". But those observers can fly away from us at speeds well in excess of c.
Am I making slightly more sense now?