There are two separate issues here:
Worldsheet versus Spacetime:
String theory is a generalization of quantum field theory, but perturbative string theory is formulated in a language pre-dating QFT. In Schwinger‘s proper time method, or “first” quantization, you construct a free field theory by summing over particle trajectories (worldlines) with an appropriate action. This summation is a path integral which formally is a “quantization” of the world-line theory. You then include interactions perturbatively by making the world-lines of those particle form more complicated graphs, which are nothing but the Feynman graphs of the corresponding field theory. This is a particularly cumbersome way to get the usual QFT perturbation theory. In this somewhat confusing language, the order in $\hbar$ is given by the topology of the graph, and has nothing to do with the process of “quantization“ of worldline theory.
(Sadly, physicists tend to call the process of solving any differential equation perturbatively “quantization”, even when it has nothing to do with QM, solving the Schrodinger equation for the time dependence of probability amplitudes, or any of that. Bad terminology is something one needs to get used to in this business.)
When generalizing to string theory, we don’t have the equivalent of the “second” quantized field theory. Instead we generalize worldlines to world-sheets, in other words point particles to strings. The sum over topologically trivial world-sheets gives classical, or free (i.e. leading order in $\hbar$) string theory, and quantum corrections (or interactions) come from including world-sheets of more complicated topology. In studying string perturbation theory it is important not to get spacetime and worksheet confused then.
In order to get free string theory you need to “first” quantize a two-dimensional field theory, which will give you the stringy generalization of a free field theory (with infinitely many fields). Free two dimensional field theory (corresponding to strings in flat spacetime) is complicated enough to require some simple sort of renormalization (e.g. normal ordering, or Polchinski’s “conformal” normal ordering). More complicated worldsheet theories (corresponding to strings in curved spacetime) require full-fledged renormalization, usually done perturbatively in the worldsheet theory as well. Note that in that case the expansion parameter is the string tension, which has nothing to do with $\hbar$, as you are still looking at classical strings which are sufficiently complicated in curved spacetime as to be usually only solvable perturbatively in $\alpha’$.
The ambiguity you noted (e.g. the normal ordering constant) is inherent in renormalization and usually one needs to impose additional physical constraints to fix these constants. In this case the ambiguity is fixed (when this is possible, e.g. with the right critical dimensional) by the requirement of preserving the symmetries of Weyl and diffeomorphism invariance of the worldsheet. Those symmetries are crucial for the spacetime interpretation of the worldsheet theory. It turns out that spacetime consistency conditions (absence of negative norm states) imply that the two dimensional field theory has to be conformal (or Weyl invariant, when formulated on curved worldsheet).
Renormalization in Spacetime
As the first part implies, this is not really relevant for this discussion. Nevertheless I want to comment that renormalization has nothing to do with the short-distance structure of the theory. The modern understanding is that it is efficient to organize QFT calculations by length (or momentum) scale, and that renormalization is the technical way to achieve that. For example, you’d want to renormalize your theory to facilitate calculations even when it is UV finite. Beyond that, probably a second semester‘s worth of QFT should clear this up.
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