I'm of the impression that "the universe is getting bigger and bigger" is the other side of the coin from "space time is curved in this really weird way". Also, I'm under the impression that a curved spacetime doesn't contradict the notion of 3D space or 4D spacetime. As I understand it, a flat spacetime is one in which space neither expands nor contracts, and convex or concave spacetime refer to space that either expands or contracts (I forget which is which). Based on observations that things are getting farther apart (or perhaps that they are accelerating away from each other?), we think spacetime has a curve.
I'm sure I'm misunderstanding something; hopefully someone can jump in and correct me.
We use a Robertson-Walker (RW) metric to describe the universe at the largest scales because it does the job well. There are several components to the (vacuum) RW metric, and two are relevant to your question.
Let's take a planar ("equatorial") slice of the expanding universe at a given time. In that slice let's put two test objects that aren't interacting with each other in any way: the gravitational attraction is effectively zero, and there are no electromagnetic or other interactions between them. They also don't decay or radiate. In the flat Minkowski spacetime of Special Relativity these test objects would follow completely parallel worldlines eternally (to the infinite past or the infinite future): the spatial distances are the same in every slice.
One parameter of the RW metric controls the spatial distance between these test objects in the immediately preceeding and immediately following slices. That is the expansion function. In an expanding or contracting RW universe, these objects are spatially closer together in one immediately neighbouring slice and farther apart in the other. In the expanding case, the spatial distances are greater in each slice into the future, and smaller in each slice into the past. "Unslicing", if these objects could (without disturbing their trajectories) measure their distances using RADAR signals, the RADAR distances would always increase into the arbitrary future. Flat spacetimes do not expand: expansion is a manifestation of spacetime curvature.
Another parameter controls whether each slice is spatially flat. If a slice is spatially curved in an expanding RW spacetime, then optical distortions change the observed size of distant objects with the expansion. In practice, this would be encoded as a distance-dependency in the brightness-angle-redshift relationship observed in distant galaxies. This isn't required by current observations made in ultra-deep-field studies, so the universe cannot depart from spatial flatness by more than a tiny amount.
The third relevant parameter is the extent of each slice. In principle every slice can be spatially infinite, no matter where in the past or future the slice is, and that is what accords with observation. However, slices could be merely finite but very large, and there might be a function relating each slice's extent to its past-predecessor or future-successor. A "closed" universe is one in which [a] the spatial curvature discussed above is positive, [b] the slices are finite but very large, and [c] there is no boundary because the slice "wraps" around spherically or toroidally or in some other fashion, and [d] the expansion function decays into a contraction function. Any non-closed universe is "open" to some extent.
This is "punned" with the non-vacuum modelling of the Friedmann-Lemaître-Robertson-Walker expanding universe with various types of matter as a fluid "dust" embedded within the vacuum RW spacetime, wherein the matter in a non-closed RW universe in the sense of the previous paragraph is dense enough that it will eventually collapse. Our two test objects above would still have always-increasing RADAR distances while all the mutually-attracting charged matter that started around them collapses into ever denser structures.
Indeed, in the FLRW model the "dust" motes are galaxy clusters, which individually collapse in a Schwarzschild-like metric (typically one uses a Lemaître-Tolman-Bondi metric, since Schwarzschild is eternal, and LTB is a collapsing dust). However, at the galaxy-cluster scales they're like our idealized test objects: they don't interact much -- after clustered galaxies form they don't really push distant ones around with their emitted radiation, and clusters are far enough apart that the mutual gravitational attraction is basically zero. Coarsely, their RADAR distances always increase into the future. (More finely, clustered galaxies orbit around inside their clusters, so some galaxies (and bits of spinning galaxies) are moving away a bit faster and some slower than expansion carries them. This is the "peculiar motion" of galaxies, and star clusters within galaxies.)
So: spacetime curvature is large, because galaxy clusters were much closer together in the past. Spatial curvature is zero or close to it, because spiral galaxies have roughly the same basic shapes to them (not squashed or stretched) at all redshifts. The universe is open in the sense that in general widely separated galaxy clusters are not at any risk of recollapsing into each other: it is only peculiar motions of galaxy clusters that cause cluster-cluster collisions, like the Bullet Cluster. (Oh, would that such collisions were commonplace: it would provide lots of useful data! But most galaxy clusters are "Eulerian": they have an unexciting view of practically all other galaxy clusters receding from them exactly according to the expansion parameter of the Robertson-Walker metric.)
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.
Roughly, here I meant far enough away for the metric expansion of space to generate significant observables. If the cosmological redshift is not evident, then the pair of events are not "mutually distant" enough.
In the "mutually distant enough" case, if the expansion is similar to that measured in our universe, then a RADAR pulse sent out by object A to object B would not return to object A before, say, about half of a sample of iron-60 at A had undergone beta decay (half-life 2.6 million years), all assuming that A and B are both moving slowly compared to the speed of light. The returning pulse will be at a significantly longer wavelength than the outgoing pulse. By contrast, a RADAR pulse that returns before half of a sample of carbon-11 has decayed by positron emission (half-life about 20 minutes), the returning pulse will be at pretty much exactly the same wavelength as the outgoing pulse. Here object A and object B move very slowly compared to the speed of light: the redshift is cosmological rather than special-relativistic.
It works for neutrinos too, which have the advantage of always moving slower than the speed of light due to their small but nonzero invariant mass. In SI units, neutrino wavelengths vary from tiny fractions of a meter to several metres. We can measure these to an extent by looking for radiation from nuclear recoil reactions: shorter-wavelength means higher momentum and thus more and stronger recoil reactions. The less famous counterpart to the cosmic microwave background -- the https://en.wikipedia.org/wiki/Cosmic_neutrino_background -- is practically undetectable in this way.
I say advantage because lightlike intervals are always zero by definition (that's why they're also called "null" intervals), so one has to use an affine parametrization of the interval to or otherwise fix coordinates and units to compare how far apart in spacetime events connected by RADAR signals are. The intervals of events connected by neutrino beams ("nadar?") are timelike, and so we can more straightforwardly consider the contribution of the cosmological constant to the (nonzero) magnitude of the interval \Delta s^2. But neutrinos are still ultra-relativistic -- simultaneously emitted neutrinos and photons (say, from extragalactic supernovae) are detected practically simultaneously by instruments on and around Earth. (In practice, such simultaneously-emitted neutrinos can win races to our detectors because the universe is generally more transparent to them than to the photons emitted from the same event: the relative opacity slows down the latter).
Sean Carroll has an excellent "zoo" of ideas in a set of slides at https://www.slideshare.net/seanmcarroll/what-we-dont-know-ab... where he gives a reasonable overview of a number of ideas including those which have an earliest time (even if it is far earlier than the hot big bang), and those which do not.
Indeed, a couple of the slides touch on Hawking's idea in the article at the top: discussing or debating that particular model (and "choosing sides") is not especially new.
The references [in square brackets] on each of the slides are mostly easy enough to find via your favourite search engine.
We are in this thing called space-time.
It came out of the Big Bang (we think).
There is a finite amount of it (about 45 GLY in radius, maybe).
You can only ever see about 14GLY of it due to some speed limits.
You can't get to an edge or boundary of space-time because it's not 3-D, it's curved in this really weird way.
There are a lot of holes in space-time that we're still thinking a lot about.
For some reason, the space-time is getting bigger and bigger, we think this is related to all that stuff in space-time, but we're really not sure.
Most of the stuff in it is not the same stuff as you are made of; most of the stuff seems to fall up, while the stuff you're made of falls down.
There is a lot more stuff that falls down than just the stuff you are made of.
There is a still a lot of things to learn about all of this, we're not really all that close to understanding it.