Let's have a vector $B_1$ which lies in the $xz$ plane forming an angle $\theta$ with $z$ axis. Let $S_\theta$ be the projection of the operator S along $B_1$.
Question 1:
How can i write $S_\theta$ as a linear combination of $S_x$ and $S_z$?
Question 2:
What is the meaning of the operator projection? The linear operator $S$ (Spin operator) is a linear transformation between two vectorial spaces, it is not a vector, how can it have a projection and components (like $S_x$, $S_y$, $S_z$)? Is like talking about the components of a function $f(x)$, what's the meaning?
I fear this is a matter of language, namely parsing out the two vector spaces involved, V and W, which are in a Cartesian product: they do not "know" about each other. V is the two dimensional plane, xz, while W is left unspecified here.
Your $\vec B_1= |B_1| (\sin\theta, \cos\theta)^T$ is strictly a vector in V, while the vector operator $\vec {\mathbf S}= ({\mathbf S}_x,{\mathbf S}_z)^T$ is a doublet of operators as written, each one of which maps vectors in W to vectors in W.
The projection is $$ \vec {\mathbf S}_\theta = \vec B_1\cdot \vec {\mathbf S}/|B_1| = \sin\theta ~ {\mathbf S}_x+ \cos\theta ~{\mathbf S}_z~, $$ now a scalar in V, and it is simply an operator mapping vectors in W to vectors in W. You've managed to decouple the two vector spaces.
A similar constructions characterizes the celebrated dot product of a direction vector to the Pauli vector which results in an operator acting on spinors (complex 2-vectors in W).