Wasn't spacetime as much curved as a black hole directly after Big Bang, because mass was so densly packed? Wasn't everything like an event horizon and how could things expand across it? Could inflation happen in a black hole too so that its content does become accessible to the outside universe?
Great question! Sorry for this huge response, and it might not be a satisfying answer, but it'll address your questions.
Sadly, as with most of astronomy, the Big Bang is surrounded in mystery. It is one of the biggest uncertainties out there, and you'll come to this realization if you've researched it enough.
Most of what we can get out of the Big Bang is through extrapolation. If we observe the expansion of the Universe, we can presume that at one point, there was far less space between all the matter in the Universe. This is where things begin to break down, though.
Through General Relativity — which explains our modern understanding of gravity — and our extrapolation, we would reach the phenomena you're describing; the entire Universe would have originated as an infinitely small point (called a singularity), from which matter was incredibly hot.
However, General Relativity begins to break down at those levels, since its equations cannot explain the conditions of such a universe accurately. For one, as you hint at, an infinite density and potentially infinite mass would likely have an infinitely strong gravity.
There is no place for such infinities in a proper mathematical description of the universe, and at that point, Einstein's equations — the basis for predicting the evolution of the cosmos — collapse. These equations provide us with unreliable results and we run into some of the problems you are describing.
Quantum mechanics is more fit for such small scales, but it fails to explain how gravity acts and what its role was at the time. Thus, this tells us that QM and GR are necessarily incomplete. Their equations cannot describe the Big Bang singularity in its entirety, especially since GR (which deals with large, high-mass phenomena) and QM (which deals with small, low-mass phenomena) are incompatible and cannot describe small, high-mass phenomena.
Thus, we're left with a lot of unanswered questions: did the Big Bang create the Universe, or was it simply an event in the Universe's history? Why would the Universe be compressed to such a small scale? Would the idea of an infinitely dense universe go against the presumption of an infinite universe?
There has long been hope that these questions can be explained within a theory of quantum gravity. Quantum gravity attempts to unify general relativity and the concepts of quantum theory (QM and QFT). These hopes have recently taken on a more concrete shape, with the help of what is called loop quantum gravity, and its applications to cosmology.
Quantum gravity is still unproven and hypothetical. String theory also tries to unify GM and QFT, but is unfalsifiable. Thus, the Big Bang is a point in time that astronomers really know little about. It's unfortunate and surprising, but it's something physicist are trying to overcome.
The Earth's gravity (call it the curvature of space) at the center of the Earth is approximately zero. Similarly, we assume a dense early universe was still infinite and homogeneous, so no worries, the gravity canceled out to approximately zero. There is no evidence that it was a lump of matter surrounded by an empty space, like a black hole usually is.
Remember, the Big Bang happened right here where you are now, and all the matter became more and more distant from the point where you are. Simply all the distances increased all the time without matter accelerating or moving into some imagined "outer space".