New question. New answer. I don't know know why you didn't ask a new question (more people are likely to look at a new question, so it can only help you).
A nonrotating black hole has a spacelike singularity, it is like a time rather than like a place. Imagine that some paths in spacetime run to the future and forever chase the surface $t=+\infty$ and their clocks can tick as much as you want and they never reach it. And nothing tremendous happens.
We call something a singularity when the curvature blows up along a curve after a finite amount of length or finite amount of time. As measured along the curve. Time along a curve and length along a curve are the real geometric facts as are the curvature invariants. The curvature blows up as the curve runs towards the future and after a finite number of ticks the curvature blows up. The theory itself fails. And this doesn't happen at the event horizon. At the event horizon everything is still normal, nothing tremendous.
And so the singularity is thought to reside within the event horizon (if it exists at all) so if you cross an event horizon it is already too late to get out to the exterior universe. So the singularity isn't a place that persists in time to be joined somewhere. It is a problem that lies in the finite future of some curves and can be avoided by others. It is not a place and so you can't be at it at one momentarily and then stay there or be somewhere else. It is like a moment stretched out over a finite region you can hit it by being in the region at that moment. But it is a moment not a place. There isn't a place to connect together.
So for a nonrotating black hole it is a moment that sits in a region inside an event horizon. Those are your parts. Outsides, event horizons, and insides that have a bad moment.
So it is not singularities that are joined together, and no one had seen a singularity and we don't even know for sure if they exist. And drawing little pictures of space alone misses the entire fact that curved spacetime curves spacetime the point is that you draw the when-wheres and then the curves between have a strange geometry that tells you how many ticks a clock makes along that cirvey (and also how much a radioactive substance decays along that curve and how many thought a person has along that curve and how many computations a computer makes along that curve and how much oxygen is consumed by a candle along that curve and in general any rate compares itself to what the weird geometry assigns to the curve).
So you can even imagine that the singularity is the surface $t=+\infty$ and that your clocks just tick so slowing along those curves that it ticks a finite number of times before you get to the end of the universe. You wouldn't talk about sewing a part of the end of the universe to a different part of the end of the universe and just saying you want to sew two singularities together is just as bad.
If you don't want to sew the singularities and instead want to sew the event horizon, the event horizon is the boundary where if curves stay on one side then the durations measured by clocks goes on and on without bound. But on the inside you are doomed to never reach that distant future because the future in front of you has curves where clocks tick a finite number of times before the end and even worse the curvature blows up in that finite number of ticks. And huge curvature hurts, like bad.
So crossing that horizon is not something you want to do. So sewing them together is bad. Because by definition the things that cross are doom and so who wants to cross.
A wormhole could be a non traversable wormhole, just two horizons that lead to the same region where the singularity is hanging in the future. Or else you can have a traversable wormhole, which generally means you have to worry about time travel. So again just drawing a constant time spatial slice of the 4d spacetime isn't going to make that clear what you are messing with.
In almost all cases when studying relativity you can benefit from studying a spacetime diagram.
A traversable wormhole either has totally different singularities, timelike ones that are like places, or else it has no singularity and has exotic matter instead. Exotic matter is interesting.
Normally if you pass through a shell of regular matter you get to a spacetime that looks like the spacetime outside a less massive star, which makes sense the shell you passed through no longer has local effects (it only affects how your clock ticks compared to clocks on the outside of the shell). But with exotic matter it can look more massive on the inside than it did on the outside before you crossed the shell of exotic matter. And even more strangely it allows totally different kinds of curved space to join up to it. So normally outside some regular matter you can have curves that focus onto the center of the earth for instance (each is pulled to the center so they each get closer) but the different curvature can instead unfocus. This allows you to be able to flare out so to get from a smaller to a bigger space as you move forwards through time. Essential when having a wormhole that opens to a bigger space as you move forward in time and the forward in time take you to the other space instead of .... to a singularity.
Removing the exotic matter is really getting in the way of going from an outside to an outside while moving through the future. I don't know what you are trying to accomplish by removing it.
