Does the spin have a spatial meaning for its direction? When we talking about direction of spin, what is a "z" direction means? For example, an electron is under a magnetic field pointing towards the ceiling of the lab, so here are the questions:

*

*Is it true that the spin states $\uparrow$ and $\downarrow$ means the spin are pointing towards the ceiling and the ground, respectively?

*Is it true that, due to the spin quantization, the spin direction can be only the superposition of pointing towards the ground or ceiling, but it cannot pointing toward the window?

*If they are true, can we measure the spin in $\uparrow$ state from the direction of window, so we get 1/2 probility of both results?

 A: the $z$-direction means whatever direction you decide to align your $z$-axis with. This is true for spins as well as for trajectory of a ball in space. The physics cannot change when someone decides to change how they label their coordinate system, and in fact we demand that the laws of physics will not depend on any specific coordinate system.
Out of convention, physicists tend to choose a basis in which the state of the spin is represented by the two $z$-aligning directions, $|{\uparrow}\rangle = |\sigma_z=\hbar/2\rangle$ and $|{\downarrow}\rangle = |\sigma_z=-\hbar/2\rangle$. However this is just a basis that we choose. It is similar to saying that we choose to describe the trajectory of a ball flying in the air using Cartesian or spherical coordinates. As long as know how to change from one basis to another, there shouldn't be any problem.
So for example, the "pointing toward the window" in your question (I guess) is the spin having definite $x$-direction, meaning $|\sigma_x=\hbar/2\rangle$. This is just another basis that we can choose, and we know how to present it using the $z$-aligning basis: $|\sigma_x=\hbar/2\rangle = (|\sigma_z=\hbar/2\rangle + |\sigma_z=-\hbar/2\rangle)/\sqrt{2}$. So really choosing a basis didn't restrict us in any sense.
As long as we didn't impose anything that will make one direction more meaningful than others, the spin in no sense is pointing "toward the cieling" or "toward the window" etc. We can break this symmetry if, for example, we apply a magnetic field in some direction, let's say toward the cieling. Now, we can call this direction $z$, $x$, or anything else (let's say $\cos(\pi/17)\hat{x}+\sin(\pi/17)\hat{y}$), the point is that now directions have more meaning. So, again to use your example, if we measure the spin in the direction parallel to the cieling-floor line, then we will only get results of the spin in that direction. But this is because we measured it that way. We did something that distinguished this direction among others. By the way, it is still important to note that we can work in any basis we want, it is just a matter of convenience.
