Does each superstring extradimension represent a (gauge) symmetry?
No. The number of dimensions required for a string theory are set by two things: demanding spacetime Poincare invariance (ie, input from special relativity) and unitarity (which for us means that physical processes are described using a conserved inner product on a positively normed Hilbert space). These are requirements of a quantum field theory as well.
In the case of local supersymmetry of the worldsheet of a string, the number of dimensions must be 10. However, as string theory was explored and fleshed out, it was found that you can consistently have 11 dimensions in a particualar phase of M-theory...M-theory is a step beyond the simple quantization of strings, where the restriction to 10 dimensions is relaxed.
Gauge symmetries (in particular, non-abelian ones) arise is various ways...one is due to the freedom to specify that the ends of an open string, e.g., transform in some representation of some group (they are "charged" with respect to some group). Compactifying or dimensionally reducing (these are different things) to lower spacetime dimensions leads to different types of gauge groups, and there are many ways of carrying these things out, including ways giving near-standard model physics.
One can also obtain 'natural' gaugings of supergravity (which can be use more directly for phenomenological purposes), which is the low energy limit of string theory. Then you can try to determine the origin of these gaugings from an underlying string theory.
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