Does fixing the reparameterisation invariance of the string action, for example by choosing the light-cone gauge
correspond to some kind of orbifolding?
This answer explains that gauge systems are orbifolds after removing the gauge redundancy. So, as the reparameterization invariance of the string action is nothing else but the worldsheet diffeomorphism invariance which is a gauge symmetry, does fixing it by the light-cone gauge correspond to some kind of orbifolding too?
And if so, what are the characteristics of this orbifold, what singularities does it have, and is there some kind of double strike which projects certain states out of the theory and leads at the same time to the emergence new ones?
If this way of thinking is wrong, I would highly appreciate any clarifications of what I am confusing.
Maybe you are confusing the worldsheet and the spacetime. What is usually called an "orbifold" in the string theory context is a spacetime orbifold. The corresponding perturbative string theory is constructed from a sigma model whose target space is an orbifold, obtained as the quotient of the action of a finite group on a 10-dimensional manifold. It is in this setting that one can obtain the states of the orbifold sigma model by restricting to the invariant states of the manifold sigma model and adding twisted states.
The diffeomorphism symmetry is a symmetry on the worldsheet. You can maybe see (very loosely) an orbifold in the gauging of the diffeomorphism symmetry as follows. The diffeomorphism group of the surface acts by pull-backs on the space of metrics on the surface. To compute a string theory amplitude, you are supposed to perform a path integration over the space of metrics modulo diffeomorphisms. While this space is the quotient of a smooth space by a group, it is in general not an orbifold in the mathematical sense of the term