The expansion of the Universe is driven by the dominant forms of energy and matter density. This is essentially an expression of a famous statement by John Wheeler (from memory, apologies if I get the exact wording wrong): "Spacetime tells matter how to move, matter tells spacetime how to curve". In this case, the matter is telling the Universe how to expand. Mathematically, Einstein's equations describing gravity, when restricted to homogeneous and isotropic Universes (relevant for cosmology), reduce to the Friedmann equations which relate the expansion rate to the matter content.
There are lots of different kinds of energy density, but for cosmology there (to zeroth order) three that really matter:
- When the Universe is dominated by non-relativistic matter (atoms and molecules but also dark matter), the size of the Universe grows with time as approximately $\sim t^{2/3}$.
- When the Universe is dominated by relativistic matter (such as photons or a very hot gas of particles moving relativistically), the Universe grows as $\sim t^{1/2}$.
- When the Universe is dominated by an approximately constant energy density (more on this in a minute), the Universe grows exponentially as $\sim e^{Ht}$, where $H$ is the expansion rate.
Note that ordinary matter (relativistic and non-relativistic) do not have a constant energy density. As the Universe expands, the energy density (energy per unit volume) goes down, because there are the same number of particles but a larger volume. (Well, it can be that the number of particles changes, but at the level of this discussion that's a detail that is not so important, the key thing is that the energy per unit volume decreases for ordinary matter).
So something weird is needed in order to provide constant energy density as the Universe expands. Physically we can imagine that there is a certain amount of energy associated with every volume of space -- then as space expands, in a sense creating more space, there is also more energy associated with that space. This is essentially the cosmological constant. Another possibility is that there is a dynamical field (a "scalar field") which permeates space, and an associated potential energy that is approximately the same anywhere.
With all that background, the current generally accepted model of cosmology is that:
- The energy density is initially dominated in some scalar field, whose large potential energy permeates space and causes exponential expansion.
- The scalar field rolls down its potential, and eventually rolls to an area of its potential where its potential energy is very small. During this process, the scalar field transfers its energy to ordinary matter and radiation fields.
- The Universe enters a phase where it is dominated by relativistic, hot matter.
- Eventually the Universe cools to a point where the matter that dominates the energy budget is non-relativistic, so the growth rate changes.
- As the Universe cools even further, and its energy density lowers, eventually the energy density of the Universe becomes dominated by a small constant value that was likely present all along, but (up until recently) was only a negligible contribution to the overall budget. This last component is what was discovered in the late 90s as the accelerated expansion of the Universe.
While this is sort of a standard picture, not all points are on equally solid footing. The first two bullet points are not universally accepted. The first bullet point is called "inflation", and (if true) explains why the Universe we see is spatially flat, and why we observe the CMB to be at the same temperature, even though the different patches that make up the CMB were not in causal contact with each other before the Big Bang in models without inflation. The inflationary period also explains the origin of the spectrum of the CMB. The second bullet point is a mysterious process that is thought to involve some combination of the decay of scalar particles into other matter ("reheating") or some dynamical process where the scalar field coherently pumps energy into other fields ("preheating").
Points 3-5, however, are on quite solid observational footing. We have observations throughout the Universe's history that probe the Universe during these different eras. While we do not fully understand what is driving the Universe today, a perfectly satisfactory model that explains all the observations is that there is a very small constant energy density that has always been pervading the Universe throughout its entire history, but we are only seeing it now because we had to wait until the Universe became dilute enough to see its effects.
Best Answer
The Big Bang is a mathematical model of how the observable universe evolved , based on fitting astrophysical observations. Like all models it has its region of validity. When I read cosmology fifty years ago, the model included a singularity at the origin, because that is what the mathematical functions of the General Relativity solutions showed. The model at the time fitted the observed expansion of the universe, which showed clusters of galaxies receding from each other at a certain rate. In that model, all (x,y,z) points at t=0 were at (0,0,0) and it makes no sense to talk outside of this point within this model, a singularity. It was as if an explosion started from the origin, in this model.
It is well known though that when singularities appear in a physics model, it is a sign for the need of an extension of the model or a different model, as happened with the introduction of quantum mechanics for the microcosm of particles, which avoids singularities with the Heisenberg uncertainty principle.
The need for a quantum mechanical framework came from the new astrophysical observations of the last sixty years. The cosmic microwave background observations could not be reconciled with classical thermodynamics within General Relativity. The CMB showed great homogenization, with differences in the temperature map of order of 2*10^-5. This homogenization could not happen at that early time of photon separation, 380.000 years in the history, and quantum mechanics was introduced at the beginning of the universe to homogenize the system.
At the same time observations showed that the expansion of the universe is not constant, a bang and free tracks, but is accelerating and new physics is introduced, dark energy , to explain the phenomenon and still keep a shell of the original Big Bang General Relativity solution.
Now to your questions:
1). As I said, everything was at one point at t=0 in the theoretical model. In the present history there is a quantum mechanical uncertainty as to the region which projects to a point from the classical BB.
2). speculations there are many. The standard physics status is in the picture above
3).is answered by my exposition above