A theta-characteristic of a Riemann surface is commonly defined as a spinor bundle, i.e., a holomorphic line bundle whose square is the canonical line bundle. These are in a natural one-to-one correspondence with spin structures on the surface, i.e., $2^{2g}$ ways to assign a winding number modulo $4\pi$ to each of homology generators $A_1,\dots,B_g$, represented as simple loops.
On the other hand, I'm assuming that the name theta-characteristic comes from the correspondence with (half-integer) characteristics of Riemann theta function. So, given a symplectic homology basis $A_1,\dots,B_g$, a choice of $\left(\begin{matrix}\varepsilon\\ \delta\end{matrix}\right)=\frac12\left(\begin{matrix}\varepsilon_1,\dots,\varepsilon_n\\ \delta_1,\dots,\delta_n\end{matrix}\right)$ with $\varepsilon_i,\delta_i\in\{0,1\}$ should give rise to a bundle as above via the theta function $\theta\left[\begin{matrix}\varepsilon\\ \delta\end{matrix}\right]$.
Q1: What is this correspondence explicitly?
It seems to be standard that if the characteristic is odd, we can construct a holomorphic section of (the corresponding?) spinor bundle as $$ \sqrt{\sum_{i=1}^{g} \partial_{z_i}\theta\left[\begin{matrix}\varepsilon\\ \delta\end{matrix}\right](0)\cdot \omega_i}. $$where $\omega_i$ are Abelian differentials of the first kind. I don't understand, though, how to figure out the corresponding spin structure. Also, if the characteristic is even, in Hejhal, Dennis A., Theta functions, kernel functions, and Abelian integrals, Mem. Am. Math. Soc. 129, 112 p. (1972). ZBL0244.30016., the Szegö kernels for the spinor bundles are constructed using theta-function. However, from Hejhal's point of view, the theta-characteristics corresponds to a character rather than spin structure, because he uses a reference spin structure provided by the Klein prime form. Figuring out the right characteristics proves cumbersome.
Q2 It appears that the Klein prime form gives, in a sense, a distinguished spin structure on a marked Riemann surface. What is it explicitly, and in which sense is it distinguished?
