How do we know the laws of physics remain the same in different dimensions? Section on Wikipedia dealing with the possibility of different dimensions.
When reading this section it feels like there's a giant elephant in the room that is not addressed. For example, here's a quote from the section:

In 1920, Paul Ehrenfest showed that if there is only one time dimension and greater than three spatial dimensions, the orbit of a planet about its Sun cannot remain stable. The same is true of a star's orbit around the center of its galaxy. Ehrenfest also showed that if there are an even number of spatial dimensions, then the different parts of a wave impulse will travel at different speeds. If there are $5+2k$ spatial dimensions, where $k$ is a whole number, then wave impulses become distorted. In 1922, Hermann Weyl showed that Maxwell's theory of electromagnetism works only with three dimensions of space and one of time. Finally, Tangherlini showed in 1963 that when there are more than three spatial dimensions, electron orbitals around nuclei cannot be stable; electrons would either fall into the nucleus or disperse.

Okay, but this assumes that in the hypothetical universe with only one time dimension and >3 spatial dimensions, the laws of physics as we know them remain valid. How do we know they remain valid? After all, if these theories were formulated in 3+1 dimensions, it can't be surprising then that they work best in 3+1 dimensions. Besides, if we're allowing the number of dimensions to vary, we're presumably also allowing the laws of physics to vary.
Concrete example: suppose it's proven that in GR, with 1 time dimension and >3 spatial dimensions, the orbit of a planet about its Sun cannot remain stable. That still isn't fatal because $f(R)$ gravity can conceivably replace GR in those universes and one of the theories might have stable orbits.
In other words, for the arguments in the quote to be convincing against extra spatial dimensions, they need to show that there is no possible theory in >3 spatial dimensions where the orbit of a planet around a Sun is stable - something which is presumably very hard if not impossible.
I'm wondering if I'm missing something.
 A: You're pointing out the weakness of any anthropic argument. Any claim that something about nature had to be the way it is, for structures like atoms or life to exist, can be countered by stating that we have insufficient imagination. This is a genuine objection, and in many cases perhaps the most important one.
However, the arguments you've cited are on more solid ground than they look. The point is that in physics, once you formulate a theory, you often automatically get theories in any dimension. For example, the postulates of Newtonian mechanics don't change between dimensions, which is why all high school courses begin with one-dimensional dynamics problems. Similarly, the postulates of quantum mechanics are totally independent of spatial dimension. In both cases the structures of the theories are rigid; it's hard to bend them without breaking them, getting something utterly alien.
In other words, mathematically these ideas seem more fundamental than $d$, which seems to be an arbitrary number tacked on at the end. That's why it's interesting and surprising to find that, in fact, if you change $d$ (while keeping the surrounding framework intact) everything falls apart. Of course this is nowhere close to an airtight mathematical proof, and it was never intended to be. Physical insights rarely are. 

If you're curious, what really powers most of the conclusions you cited is special relativity. The geometrical structure of special relativity follows by postulating a spacetime metric
$$\eta = \text{diag}(-1, 1, 1, 1).$$
To accommodate more spatial dimensions, you just add more ones at the end. To accommodate more time dimensions, you just add more $-1$'s at the beginning. So then you can ask, assuming the postulates of special relativity hold, would physics work if the number of dimensions were changed in this way? Again, this is an interesting question to ask if you treat relativity as a deeper fact that the number of dimensions.
It turns out relativity is so powerful that we get a lot of mileage out of changing $\eta$ alone. For example, the very simplest equation of motion for a field is 
$$\eta^{\mu\nu} \partial_\mu \partial_\nu \phi = 0$$
which is called the wave equation, and describes fields propagating at the speed of light. So you can analyze the wave equation in an arbitrary number of dimensions, which is what was done in the second equation you cited. The wave equation also can be used to derive how fields fall off with distance (e.g. inverse square in 3 spatial dimensions), allowing one to analyze orbits in arbitrarily many dimensions, either classically (your first example) or in quantum mechanics (your fourth example). Your second example is a little weaker; Weyl showed that electromagnetism only has a certain cute theoretical property, conformal invariance, in $3$ spatial dimensions. But this property can again be determined solely from the structure of special relativity.
