Light-cone quantization of open string as derived in Polchinski Polchinski uses the following gauge conditions, but I don't follow this procedure of gauge fixing and quantization:
\begin{align}
X^+ = \tau, \tag{1.3.8a} \\
\partial_\sigma \gamma_{\sigma \sigma} = 0,\tag{1.3.8b}\\
\det{\gamma_{ab}} = -1.\tag{1.3.8c}
\end{align}
Please let me know if you could break it down.
Secondly, Polchinski says that the classical theory is Lorentz invariant for any $D$, but there is an anomaly - the symmetry is not preserved by quantization procedure except when $D = 26$. I understand the later part; but classically, does he just mean the the Lagrangian we started with was Lorentz invariant in any $D$? But even for Polyakov action classically, we had to use constraints that removes manifest Lorentz covariance?
 A: Your quote, "we had to use constraints that remove manifest Lorentz covariance," makes no sense. The constraints of the Polyakov action do not break Lorentz symmetry since the Polyakov action is Lorentz invariant. Solving the constraints by a given set of variables, say light-cone gauge, will make the Lorentz symmetry non-manifest, i.e., realized non-linearly, with non-covariant expressions. However, the symmetry is still there classically.
Quantization will generically break this symmetry due to an anomaly, an obstruction in asking for many symmetries: reparametrization invariance + Weyl symmetry + Lorentz symmetry. Since the light-cone gauge from the Polyakov action requires the usage of reparametrization invariance and Weyl symmetry, the symmetry that will cause problems under light-cone quantization is Lorentz symmetry.
Indeed the guiding principle in light-cone quantization of the (super)string theory is the preservation of Lorentz symmetry, which fixes the interaction altogether.
In different gauges, say conformal gauge, the Lorentz symmetry is manifest, so quantizing while preserving this symmetry will be easy. However, Weyl symmetry (or reparametrization, depending on how you formulate conformal symmetry) will be problematic under the worldsheet quantization, and preservation of conformal symmetry will be the guiding principle to develop interactions, similar to Lorentz symmetry for the light-cone quantization.
The conformal gauge has the advantage over the light-cone gauge since Lorentz symmetry is manifest. For example, in light-cone quantization of superstrings, one should perform insertion at the interaction points of the string, points in which the strings split and join, to maintain Lorentz symmetry. Unfortunately, it isn't easy to calculate the location of these points.
In bosonic string theory, one can get away with this problem since the insertion can be ignored by a conformal map from the light-cone worldsheet to a smooth Riemann surface, so the insertion becomes trivial. However, for superstrings, the insertion is still non-trivial after the map.
Interestingly enough, the traditional Lorentz covariant formalism for the superstring, Ramond-Neveu-Schwarz formalism (chapter 10 of Polchinski), suffers from not maintaining space-time supersymmetry manifest, which causes some problems such as the lack of a worldsheet action for backgrounds with non-zero Ramond-Ramond fluxes. This is the main motivation for the search for other worldsheet formulations of the superstring theory, such as pure spinor formalism.
