The hot big bang is an event in spacetime. It does not generate the manifold itself, although the geometry of the manifold necessarily reflects it, especially near the event itself.

There is no evidence to support finiteness of spacetime; there is no reason why there aren't events in the infinite future. The only reason to suspect there are no events in the infinite past is a classical picture of a gravitational singularity in the finite past, but we good reason to believe that quantum gravity will become important in the finite past, and that quantum effects prevent the singularity from forming. However, we do not have a trustworthy theory of quantum gravity with which to assess a number of ideas about how one might test predictions about the even more distant past.

Light-years are a measure of spatial distance; in spacetime we must use an interval for several reasons, including that different observers will disagree about the amount of time it takes a pulse of light at A to reach B; the distance will vary depending on where in spacetime observers are, and the geometry of the spacetime. In order to be generally covariant, intervals must be tensors. We can write the interval tensor in a linear form as \Delta s^2 = x^{\mu}x_{\mu} where \mu is an index of spacetime dimensions runnning 0, 1, ... depending on how many of them there are, x is a displacement four-vector (covariant with \mu below, contravariant with \mu above), and the whole right-hand-side is a Minkowski inner product. The interval itself is s^2; it is not the square root of this quantity.

If we discard general covariance by fixing flat polar coordinates on one observer, we gain the ability to discuss light-years (as measured at some point in time at the spatial origin in that coordinate basis) but invite mistakes in relating those units to physical systems. One runs into this a lot on hackernews, where someone inevitably resurrects the objection that e.g. a binary black hole merger detected at LIGO today akshually happened billions of years ago, and in the process makes a complete hash of the metric tensor.

Unfortunately, this is what is happening in your line about the time-dependent Earth-fixed Hubble radius and the line immediately after that. The metric, as you say in the very next line after that, is very far from that of flat spacetime, and light-years become tricky in general curved spacetime.

As said above, there may be an early boundary to the spacetime: there might not be an infinite past. Not all singularity-abolishing ideas involving quantum gravity require the extension of spacetime beyond the hottest densest phase of the universe, and not all extensions must be infinite. There may be a future boundary to the spacetime, but evidence is that the true metric (which we do not fully know; we only approximate it with an expanding Robertson-Walker metric in the standard model of cosmology) extends into the infinite future barring possible quantum gravity effects at extremely low energies.

> most of the stuff seems to fall up

No. Dark energy is not "stuff": stuff dilutes away with the expansion, and locally tends to slosh about in response to gravitation. Evidence supports the assertion that dark energy is a (physicists') choice of how to represent a linear element in the spacetime interval between any pair of mutually-distant events. That element is \Lambda, the cosmological constant.

As far as we can tell the gravitational interaction is only attractive for all "stuff"; the dilution only manifests when the gravitational interaction is extremely weak. The attractive interaction in the Friedman-Lemaître-Robertson-Walker model in the standard cosmology is represented as a pressure in a fluid dust of gravitating matter; it is calculationally convenient to represent the cosmological constant as a constant isotropic tension. However, the convenience comes with similar risk of misunderstandings, like with using light-years to talk about the intervals between two events separated by cosmological distances.

> a lot more stuff that falls down than just the stuff you are made of

One of the features of galactic halos is that they do not "fall down" towards the central parts of the galaxies they enclose. Ordinary matter collides or scatters electromagnetically or through the weak nuclear force, and and such scatterings radiate away momentum as photons, neutrinos, or other particles. The reduction in momentum allows the remaining matter to fall inward. The matter in the halo does not produce photons, and does not seem to produce neutrinos, so is effectively unable to fall inwards (except by dynamical friction, which is an extremely slow process for sparse gasses or dusts of small-mass particles).

Additionally, we cannot be certain that dark matter -- if it interacts non-gravitationally -- does not form bound states with ordinary matter such as the atomic nuclei inside our bodies. One could compare this to the "hot dark matter" neutrinos that are in your body at any given moment thanks to nuclear interactions (for example, in beta decays in the potassium in your blood). ("hot" because neutrinos generally move at speeds close to that of light; "cold dark matter", found in halos, moves much more slowly).

> There is still a lot of things to learn about all of this

Yes, you're right here.

> we're not really all that close to understanding it

But here I think you are wrong, if your use of "we" is meant to include working physical cosmologists.

Tying those two together, I'd happily recommend practically any of these https://en.wikipedia.org/wiki/Physical_cosmology#Textbooks