Can separate manifold regions have the same coordinates? Despite referring to physics, this question is purely mathematical on geometry and topology of the given pseudo-Riemannian manifold.

Case A
As a starting point, consider the Schwarzschild solution extended to the interior of the horizon. Here we have two regions, external and internal with known different properties.
We can describe the global spacetime (disregarding the maximal Kruskal extension) by two regions of the Schwarzschild coordinate chart, internal and external. (In this question, I am not concerned with the horizon boundary.)

Case B
Now consider a thin, hollow, massive shell of dust collapsing under a spherical symmetry to its Schwarzschild radius. As known, the spacetime outside the shell is Schwarzschild while the spacetime inside is time dilated Minkowski.
We can describe this spacetime by two regions of a chart in the Schwarzschild coordinates. So far, so good.

Problem
A trouble begins in the frame of the collapsing shell where the proper time to the Schwarzschild radius is finite. There we have two logical options:


*

*The proper time of the shell ends at the horizon. In this case, the above mentioned coordinate chart (Case B) covers the entire global spacetime manifold (short of the horizon boundary perhaps).


*The shell in its frame continues through the horizon. Then, per the Penrose Singularity Theorem, a singularity must form on the inside. In this case the spacetime there should look somewhat similar to the Schwarzschild extension (Case A).


In this case we end up with two spacetime regions on the inside: one is time dilated Minkowski (before crossing the Schwarzschild radius in the shell proper time) and the other is similar to extended Schwarzschild with a singularity (after the crossing). They seem to overlap over the same set of the Schwarzschild coordinates: for any $t,r,\phi,\theta$ there exist two different events in these spacetimes.

Question

Can two different regions of the same global spacetime manifold share the same coordinates?


Sorry for the lack of rigor, I am not a mathematician. If any clarification or additional conditions are required, I would be happy to provide. Please do not hesitate to ask. Thanks for your expert insight!




EDIT - Based on comments, below is a clarification on the terminology used in the question. Nothing new here, just some optional background for clarity.

Schwarzschild
The Schwarzschild spacetime is a pseudo-Riemanian manifold defined by the following metric in polar coordinates:
$$ -{d \tau}^{2} = -\left(1 - \frac{r_\mathrm{s}}{r} \right) \,dt^2 + \left(1-\frac{r_\mathrm{s}}{r}\right)^{-1} \,dr^2 + r^2 d\Theta^2 $$
Where $d\Theta^2$ denotes the spherical metric induced by the Euclidean on a two sphere, i.e. 
$$ d\Theta^2 = d\theta^2 + \sin^2\theta \, d\varphi^2\;\;\; \text{and} \;\;\; r=\sqrt{x_1^2 + x_2^2 + x_3^2} $$
Here $r=r_s$ is a sphere of the event horizon, a coordinate singularity where the temporal part of the metric is zero while the spatial part radially diverges. The Schwarzschild metric accurately describes the gravitational field outside an uncharged non-rotating spherical object, such as a planet, star, black hole (or a hollow spherical shell in this question).
By extending this metric through the horizon, we notice that $t$ becomes spacelike while $r$ becomes timelike on the inside. This extension is the mainstream interpretation of the spacetime geometry inside an uncharged non-rotating black hole.
It is easy to see that a timeslice inside the horizon in the Schwarzschild coordinates is a spherinder rapidly shrnking in time $r$ to its axis (along $t$) called the Schwarzschild singularity, an infinite line $(r=0,-\infty<t<+\infty)$ removed from the spacetime manifold: Is the Schwarzschild singularity stretched in space as a straight line?
In this question, in the Case A, the Schwarzschild metric applies both outside and inside the horizon; in the Case B, this metric applies only outside of the massive spherical shell.

Minkowski
The Minkowski spacetime in a hollow massive shell is a flat pseudo-Euclidean manifold defined by the following metric (where $t\equiv x_0$):
$$ -{d \tau}^{2} = -H\, dx_0^2 + dx_1^2 + dx_2 + dx_3^2 $$
or in polar coordinates:
$$ -{d\tau}^2=-H\,dt^2+dr^2+r^2d\Theta^2 $$
See Weinberg, "Gravitation and Cosmology", p. 337 where $H$ is denoted as $f(t)$.
Here $H$ defines the time dilation (squared) and can be renormalized to unity in the coordinates inside the shell, but not in the Schwarzschild coordinates, because $dt$ must be continuous through the shell (the same time dilation inside as at the shell):
$$ H=1-\dfrac{r_s}{R} $$
where $R$ represents the radius of the massive shell, so the time dilation is the same anywhere inside the shell, at any radial coordinate $r$. See: On a Common Misunderstanding of the Birkhoff Theorem where $H$ is denoted as $h(t)$.
In this question, this metric applies to the Case B inside the massive spherical shell while the shell is larger than the horizon $r>r_s$ (which is forever in the Schwarzschild coordinates).
 A: It is not easy at all to try to guess what would happen without setting up the problem mathematically and trying to solve it.
If you are interested in the problem of a massive shell of dust collapsing, and the resulting spacetime geometry, well you have to set up the problem, namely write down the stress-energy tensor for this problem and try to solve Einstein's equations.
This is probably too difficult to attack by brute force, so to speak. I would try to look first at what has been done in the literature on that problem and possibly similar problems.
Especially in an area like GR, it is not at all clear how to guess what the solution metric looks like, even qualitatively (at least not for me!).
I think this is the main problem you are interested in. I did not provide an answer (as it is a research project on its own), but I did provide some guidelines that would hopefully be useful to you.
Regarding the mathematical language, local coordinates are just numbers to describe the local position of a point. You can have different sets of local coordinates to describe the same local region on a manifold. It is a bit like using two different maps to describe locations on the surface of the earth. In flat space, you could for instance use coordinates coming from an orthonomal coordinate system, but you could also for instance use spherical coordinates. The same metric would look very different in $2$ different coordinate systems, but if you inspect things closely, you would notice that the intrinsic properties are the same. It is like having two different descriptions of the same space. This leads to the notion of two metrics being isometric.
Edit 1: It seems that what safesphere is really interested in, is the notion of a covering space. The OP's questions can be subdivided into two categories: topological and Riemannian.
Let us start with the topological side. Without getting too much into technicalities, let us just say that covering spaces are closely related to the notion of fundamental group. There are known sufficient conditions for the existence of covering spaces, but let us just say that these conditions are satisfied if the base space (the space you are trying to build a covering space of) is a connected topological manifold. If the base manifold is further simply connected, there aren't any "interesting" covering spaces, while if the base manifold is not simply connected, there are some "interesting" covering spaces. I am being vague and sweeping a lot under the rug. For details, you can look at Munkres's Topology for instance, or Hatcher's book on topology (which was free to download at some point).
An example of an interesting covering space, is $SU(2)$, which is diffeomorphic to the $3$-sphere, and is a $2$ to $1$ covering space of $SO(3)$, itself diffemorphic to the $3$-sphere with antipodal points identified (thus diffeomorphic to real projective $3$-space).
There are more general types of covering spaces, called branched covering spaces, which occur naturally in algebraic geometry. This makes answering your topological questions more complicated.
Working out the topology of the covering space is usually not difficult in specific problems, where the topology of the base space is known. Let us just say that the topology of a covering space is closely related to that of the base space. It is in some sense the topology of the base space, but unwrapped a number of times (the number of times possibly being infinite).
Here is an example of an interesting covering space. The real line $\mathbb{R}$ is a covering space of the circle $S^1$, thought of as the unit circle in the complex plane $\mathbb{C}$. Indeed, the map $p : \mathbb{R} \to S^1$ defined by $p(t) = e^{2 \pi i t}$ is a covering map. In this case $p(t+n) = p(t)$ for all $t \in \mathbb{R}$ and all $n \in \mathbb{Z}$.
As to the Riemannian part of your question, a covering map in the Riemannian sense, from one Riemannian manifold to another, is first of all a covering map in the topological sense, such that the pullback of the metric on the base manifold is the metric on the covering manifold.
So for covering maps in the Riemannian sense, "above" a (small enough) local neighborhood of a point on the base manifold, lie a disconnected union of isometric copies of that neighborhood. In other words, the copies in the covering manifold are isometric (look the same as) to the local neighborhood in the base manifold.
You may then ask: can't we, say, glue two copies of two different manifolds. You can, but then you lose the notion of a covering. It will be a topological gluing construction. You take two manifolds and glue them together, basically. I don't think this is what the OP really wants to do. Moreover, if you want to glue two Riemannian manifolds, you will have to make sure that the metric is smooth even where you do the gluing. This is in general very difficult (and is often impossible) to satisfy.
Now can we have a metric on a covering space (in the topological sense) which is not related to the metric on the base manifold? You could of course, but in practice, the natural metric on a covering space is the pullback metric (which thus looks locally like the metric on the base manifold).
It is difficult to answer your $4$ questions more precisely, as the answer will depend on how you translate them into precise mathematical statements. What I mean is: are you asking about covering spaces, or perhaps more general branched covering spaces?
I will just say that for covering spaces, usually the topology and (natural) metrics on the covering manifold are very closely related to those of the base manifold. However, for branched covering spaces, the situation is more complicated due to the existence of a branched locus.
A: This is not a real answer, but an extended comment. In my opinion there is a big problem: We have the language of physics and the language of  mathematics, but we do no have a good interpreter. I am sure that good interpreters exist, but I do not belong to them. Perhaps you should talk to mathematicians at your univerisity, an intensive dialog is much better than a conversation in a forum.
That being said, I think the situation is this:
You have two observers watching the same spacetime region. In the language of manifolds, this region is an open subset $U$ of the spacetime manifold $M$. Depending on their position they see different things, formally they use different local coordinate systems for $U$. A local coordinate system is a homeomorphism $\phi : U \to V$, where $V$ is an open subset of the standard Euclidean space $\mathbb R^4$. There are infinitely many such local coordinate systems for $U$. As a simple example in dimension $1$ take the set $S = \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 =1, x, y > 0 \}$ . This is an open quarter circle and the maps $f : (0,\pi/2) \to S, f(t) = (\cos t, \sin t)$, and $g: (0,1) \to S, g(t) = (t, \sqrt{1-t^2})$, are homeomorphisms. Their inverses are local coordinate systems for $S$. The first of them describes what an observer sees from the origin $(0,0)$, the second what an observer sees from a distant point $(0,R)$ with $R >> 1$. Now consider a point moving couterclockwise with constant speed along $S$ in direction $(0,1)$. The first observer sees this point moving with constant speed, but the second observer sees that the speed goes to $0$ as the point approaches $(0,1)$.
