Bravo! You have put your finger on several deep questions!
First, what is the geometry ("curvature") of the universe, and how do we know it? Einstein's theory of gravity, General Relativity, allows for the three-dimensional space of the universe to have one of three possible shapes or geometries:
(a) space can be "positively curved"--like the surface of a expanding balloon
[but a 3-D upgrade of this idea!] and with an finite volume, or
(b) space can be "negatively curved"--like the seat of a saddle or a potato chip
[again upgraded to 3-D] and with an infinite volume, or
(c) space can be "flat"--lacking any curvature at all and infinite in volume.
In each of these situations, geometry is different, meaning that the properties of circles and triangles are different. For example, the familiar Euclidean results about triangle interior angles summing to 180 degrees, or circle areas being pi*r^2, are only true in a "flat" universe and are false in the curved universes! More discussion and illustration appears here: http://background.uchicago.edu/~whu/beginners/curvature.html
Thus Einstein teaches that geometry is really a realm of physics, and it is an experimental
task to measure which of these is the true geometry of the universe. And in fact, this experiment has been performed, using observations of the cosmic microwave background, which are described here http://map.gsfc.nasa.gov/universe/uni_shape.html http://map.gsfc.nasa.gov/media/030639/index.html
and ultimately amount to studying the geometry of huge triangles in the universe!
And the answer is: the universe shows no average curvature but rather has the "flat," Euclidean geometry on a large scale, within the error bars of the measurement.
Another question you ask is: can this geometry be different elsewhere in the universe?
While in principle this is possible, all of the available evidence agrees with another prediction that goes back to Einstein: the universe is homogeneous, meaning that at
any time its large-scale properties (including geometry) are the same at
all points in space. Thus if the observable universe (the amount anyone can
see since the big bang) is flat, the the entire universe is as well.
Finally, another question you raise is: what properties does the universe have
at exactly t=0, the moment of the big bang? This situation is known as a "singularity"
which is the technical way of saying that the known laws of physics
break down. This means that nobody knows what happens then! But
many ideas exist, in which people try to extend known physics to this extreme regime
by inventing theories of quantum gravity. If you have a successful theory
of quantum gravity yourself, you should publish!
(published on 07/04/2011)