On how the math works....
Can you explain this? What does "something is allowed by the math" mean?
Morrus and Umbran gave some great initial answers, so I'm just going to give those a shout-out here rather than quote them (given the length this post will be).
In fact, I think rather than quote much of the above posts, I'm going to condense some of the points/questions and offer my thoughts. My apologies if I mangle what you're trying to say, but I want to paraphrase and go a bit out of order. I'll highlight when I start talking about a new point.
One conversation thread was basically Bullgrit asking about the math corresponding to a wormhole and how/why physicists came up with it.
First off, we can talk about the "amount" of math needed. At the basic level, it's easy to write down in one line a formula that, with training, you could look at and puzzle out "oh, that's a tunnel connecting two parts of space." However, then you want to learn about it. That's where you can get into reams of paper. And
Bullgrit said:
I mean, I'm picturing someone looking at those whiteboards of equations, saying, "Hmm, what if this variable right here was a 2?" And the whole equation calculating out three feet away to "= tiny spacetime tubes connecting two points".
is great, because there are times like that for any theoretical physicist. Well, it's usually some symbol rather than a 2, and usually you're not changing things at whim but trying to figure out new ways to rearrange them, but, yes, I sometimes make a point to work on my chalkboard rather than paper just because the visual effect helps.
As to "why wormholes," my understanding is that people went looking for them. Let me explain more precisely. The Einstein equations of general relativity take in sketch form "geometry" = "matter-energy." Usually, what we do is start with some configuration of matter-energy and ask what the corresponding geometry of spacetime is. That's not what people have done in the case of wormholes. People wanted to study a specific geometry, namely a tunnel between two regions of space, so they plugged that into the Einstein equations and figured out what kind of matter-energy is required. That's why we've been talking about "exotic matter" so much in this thread. You'd never find a wormhole if you were looking at all the ways normal matter than bend spacetime. In a way, wormholes were a solution looking for a problem. I suspect that sci-fi fandom may have played a role motivating people to look at wormhole solutions, but I don't know that history.
On math as a language of physics (or maybe all science, if you like), Umbran states very nicely just how powerful math is in describing the world (yes, it is awesome). My personal view is a bit more along the lines that mathematical principles are something we discover, like we discover the laws of physics, but that's really more like a feeling.
If you want to
compare wormholes to dark matter, you have two very different situations. As I said, people really went looking for wormholes. On the other hand, basically no one believed the first evidence of dark matter for decades until a new set of evidence came along. So we were really sort of forced to admit that we needed to add something to our models of cosmology. Interestingly, there's not really new math required --- dark matter behaves according to similar principles as normal matter in the simplest theories.
But dark matter is predictive: once we knew that we needed dark matter to explain the rotation of galaxies, we could predict the consequences of that amount of dark matter for the cosmic microwave background (light from the very early universe), and the predictions really match the measurements excellently. We always like predictions out of physics theories.
Regarding the history of special/general relativity and quantum mechanics, Einstein was very much thinking about several unexplained experiments when he discovered special relativity, even though the thought experiments look unrelated. He had that rare genius to relate that simple thought to problems that were confusing everyone else. On the other hand, he did discover general relativity just because of questions in his own mind. There were no experimental prompts for him. But a successful prediction of general relativity was what made him famous. As for quantum mechanics, the basics were again discovered in an effort to understand confusing experiments, but, as Umbran says, there were indeed predictions of the math that weren't seen experimentally for a long time.
But also remember that all the discussion of wormholes in the literature (at least that I've seen) is about wormholes that exist forever unchanging or else analysis of whether a wormhole that's been around forever collapses if something goes through it. There's nothing at all about the formation of them. (This is really quite different than the situation for black holes; I'm writing a series of papers about black hole formation myself, for example.) 
