Regarding the topology of the universe, it could be compact like a sphere or open like a Euclidean space, but since the universe started from a single point, doesn't that mean that the shape of the universe must be contractible?
Compact manifolds are not contractible, so how could the universe be compact?
The main point is: don't make the mistake of extending classical GR all the way to a singular point! In sufficiently extreme conditions the physics of the universe is not known, but since quantum physics is involved, it is likely that no classical manifold is adequate to describe the early universe.
In big bang cosmologies with an initial singularity, spacetime only exists at $t>0$. Spacetime manifolds don't actually have singular points, they just have open edges beyond which they can't be extended. I'm not sure that the spatial topology at $t=0$ is a well defined concept, but the horizon problem suggests that it shouldn't be viewed as a single point in general. Cosmologies with positive spatial curvature can have a horizon problem, so even a finite total spatial volume that goes to zero as $t\to 0$ isn't enough to guarantee that the initial singularity can be reasonably treated as a point.
In cosmologies (probably based on quantum gravity) that really do start at a nonsingular point, there is no obvious obstacle to having noncontractible spatial slices at later times. Take the Euclidean plane with cosmological time being distance from the origin. Space is a point at $t=0$ and a circle at all $t>0$. This example has the wrong metric signature, but there is no obvious reason why a real quantum-gravitational cosmology couldn't have broadly similar features.
I am unfamiliar with the usage "compact manifold", and the clearest definition I could find is "A compact manifold is a manifold which is compact as a topological space," in https://topospaces.subwiki.org/wiki/Compact_manifold . Since this was not helpful, I then found that the terminology is quite ambiguous in https://topospaces.subwiki.org/wiki/Compact_space . The clearest definition example there is: "For a metric space to be compact with the induced topology is equivalent to a condition on it called being totally bounded."
OK then, with this definition it seems that a compact universe is a finite space with a boundary. However, a cosmological model of a finite universe has no boundary. So, the vocabulary used in the question you are asking seems to not be applicable to cosmological models.
