I am not a GR expert, so I cannot give answers with any authority, and I won't try to match the level of rigor of the preceding entries. However, I hear one question in what ngeo is saying that is easier to answer than to use the full GR formalism. I think I'm hearing "what is causing the universe to expand", in hopes that it is contained in the G=T equation. However (and please correct me if I'm wrong), the G=T equation has rather the flavor of Newton's equation for acceleration, i.e., acceleration equals force divided by mass. If you apply that equation to a cannon ball, and ask, why is the cannon ball flying through the air, you see that the answer is not contained in the equation. This is a very important point about the Big Bang, I think, that is not at all widely appreciated. The cannon ball is flying through the air because it was shot from a cannon, not because of Newton's equation. It is just an initial condition, whose subsequent evolution is governed by Newton. You could imagine a very different governing equation, yet cannons could still shoot balls. When applied to the Big Bang, this means that the most important thing about the universe is the expansion, which arises not from GR but from a presently ad hoc initial condition. You could have other theories of gravity, hypothetically, and you should expect that a universe that was originally expanding might still be, given the right conditions. Worse, since we now have to postulate both dark matter and dark energy, both unknown, to make things work out, I would argue that the only element of GR that has anything directly to say about whether or not the Big Bang is an accurate phenomonelogical description, is the fundamental instability in the structure of GR. That is, an unstable force principle should lead to a dynamical result, and the Big Bang is indeed dynamical. Beyond that, it's all details.
In short, GR works because of its independent verification in our own backyard, not because of anything that has to do with cosmology. Since whenever we see a glitch in the joint application of GR and the observed expansion, we merely postulate new physics, I would claim that
GR and the Big Bang are completely independent theories (both on very solid footing, in terms of their usefulness as organizing principles).