Why are superstring models in physics needed

Let me take parts 2. and 3. of the question first:

The 10 dimensions of string theory are, a priori, not "coiled up" or anything else. They are derived for a string theory where the classical version of the string propagates in d-1 spatial dimensions and 1 temporal dimension, i.e. Minkowski space $\mathbb{R}^{1,d-1}$. "Dimension" here is dimension of a manifold in the usual sense of differential geometry - number of coordinates needed to uniquely distinguish a point on the manifold from all points close to it.

Now, as for why (super)string theory in flat space requires $d=10$:

One way to see string theory is by certain two-dimensional conformal field theories living on the world sheet the string traces out in the target space. I give a quick explanation of the structure of such theories here. The total conformal charge of the full combined CFT on the worldsheet can be seen as the quantum anomaly of the classical Weyl symmetry of the string - for a general discussion of the relation between anomalies and central charges see this answer by DavidBarMoshe, for a general discussion of the relation between central charges and quantization see this Q&A of mine.

The quantization of the bosonic (or "naive") string has d coordinate fields that each correspond to a free bosonic CFT with central charge $c=1$ plus a "ghost system" incurred from BRST quantization that has a central charge $c=-26$. Ghost systems are allowed to have negative central charge because they decouple from all physical processes.

Now, the procedure used to quantize this string in the first place makes use of the Weyl symmetry being non-anamalous, i.e. $c=0$ for the full theory - which only happens at $d\cdot 1 - 26 = 0$, i.e. $d=26$. Therefore, the bosonic string exists consistently as a quantum theory only in 26 dimensions.

The superstring is now what you get when you additionally have fermions living on the worldsheet. It's called the "super"string because the new action is supersymmetric, but it might as well be called the "spinning string", since trying to write down a worldline action for a particle with spin also introduces such fermions.

In any case, the ghost system for the larger symmetry of the superstring has $c=-15$, and the fermions each contribute $c=1/2$. This gives the requirements $\frac{3}{2}d - 15 = 0$, which is solved by $d=10$.

I'm afraid the full derivation is rather technical and it would serve little use to reproduce it here. Lastly, one should remark that there are many equivalent ways to arrive at this constraint on dimensions, this is by far not the only one, but the one that's easiest to tell for me. Others might find a presentation discussing ordering constants related to the vacuum energy more physically intuitive, for example.

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