The Schwarzschild solution is the unique distribution in GR for nonrotating mass in a small area, in a universe that is asymptotically flat at a long distance from the mass. This is not a flat universe, but most of it is pretty darned close to flat.
As for describing the shape of space-time, that's what GR does. What we can think of as the "shape" is actually described by something called the metric. GR says that the metric satisfies a differential equation. If the universe starts close to flat, things are moving slowly, and there is a low density of mass, the solutions to this equation create an effect that, to first order, matches Newtonian gravity. But the full theory has solutions with all sorts of bizarre things in it, like waves traveling through space, made up of fluctuations in the very structure of space-time. We call those gravity waves.
And yes, those solutions do include things like expanding universes. And the effect of gravity within an expanding universe is to slow the rate of expansion.
The metric you're referring to, oddly enough is a mapping from flat spacetime to curved. This is why the Schwarzschild and Kerr solutions to the EFE have `r` values that are in flat spacetime and yield spacetime intervals. The metric is symmetric, so you can also map back from curved to flat spacetime.
As for describing the shape of space-time, that's what GR does. What we can think of as the "shape" is actually described by something called the metric. GR says that the metric satisfies a differential equation. If the universe starts close to flat, things are moving slowly, and there is a low density of mass, the solutions to this equation create an effect that, to first order, matches Newtonian gravity. But the full theory has solutions with all sorts of bizarre things in it, like waves traveling through space, made up of fluctuations in the very structure of space-time. We call those gravity waves.
And yes, those solutions do include things like expanding universes. And the effect of gravity within an expanding universe is to slow the rate of expansion.