Definition of spacetime in GR

A manifold is a set - you don't need to put the manifold structure onto anything. Take a look at the first line of the wikipedia page for a manifold: a manifold is defined as a topological space which satisfies certain properties (and a topological space is a set of points).

Intuitively: a manifold is a set which looks flat if you zoom in close enough on any of its points. This is where your notion of $\mathbb{R}^4$ comes in - since any spacetime in GR is a manifold, this means it looks like flat Minkowski space ($\mathbb{R}^4$) if you zoom in close enough on any of its points.

If you zoom out, the space may be curved and not resemble $\mathbb{R}^4$ at all (as is the case for Schwarzschild). The thing you use $\mathbb{R}^4$ for when describing Schwarzschild space is to describe points on the manifold with coordinates (see coordinate chart on wikipedia).

Coordinate charts always can map/describe a region $U \subseteq \mathcal{M}$ of a manifold $\mathcal{M}$, but sometimes they fail to describe the entire manifold (ie. sometimes $U \neq \mathcal{M}$ for a particular coordinate chart). Or the coordinates might have singularities at certain points on the manifold as well (as is the case in Schwarzschild space: ordinary Schwarzschild coordinates famously break down at the horizon).

The set is not predetermined but arises from physical/mathematical requirements of the given solution.

GR is local theory and sufficinetly small region of spacetime is assumed to be isomorphic to open region of $\mathbb{R}^4.$ Globally the set is given by "gluing" these regions together until you arive at global solution you are satisfied with. GR does not strictly speaking enforce this. However, it is reasonable to demand some properties like smoothness of the metric, maximal extension and so on. In the case of Schwarzschild spacetime, these requirements are strong enough to guarantee uniqueness.

Mathematical conditions such as Hausdorf or paracompactness apply to mathematical models of reality. They are introduced to prove theorems that apply to these models. Do not confuse mathematical models of reality with reality itself. Whether the universe is everywhere Hausdorf or paracompact is something to be decided by experiment. No amount of studying the continuum hypothesis or alternative axiom systems for the real line can tell us anything about the space in which we live.