Direction of normal force on stick on box 
What is the direction of the normal force on the stick in this case, assuming gravity? Is it right angled with the stick? Or is it upwards? Or is it impossible to determine?
 A: Normal is a synonym for perpendicular. 
The normal force is as you show it, perpendicular to the slanted object. Gravity is a separate force having a different agent (the earth) and plays no role in determining the direction of the normal force. Friction is parallel to the surface, and  is not a normal force.
A: I think the problem is not the stick, it is the edge of the box. For an ideal box the edge changes direction discontinuously : one side is vertical, the other is horizontal. So how can the normal to the surface of the box at an edge be anything other than horizontal or vertical?
For a real box the edges cannot change direction discontinuously. If we look on a small enough scale an edge changes direction gradually, and can be approximated by a section of a continuous curve, such as a circle. Then it is easy to identify a point of contact at which the surfaces of the stick and box are parallel, and a common direction which is normal to both.   

However a continuously changing surface is really yet another idealisation. On a microscopic scale most real contacting surfaces are jagged an irregular, and deform in response to the forces between them. The result is that there are many individual points of contact at different angles, and many different contact forces. The macroscopic contact force is the sum of these, and its resolution into parallel and perpendicular components is a convenience for mathematical analysis of the situation.
A: There are three scenarios here. First consider the equations of motion



*

*no friction - The contact force is normal to the contact (direction $\hat{n}$) only. The body will slide along the contact plane (direction $\hat{e}$). The magnitude of the contact force is that there is no motion of the contact point along the normal direction.

*slipping, low friction - There are two components to the contact force. One is the same as above along $\hat{n}$ and one is frictional along $\hat{e}$. The relationship between the two is $F = \mu_{\rm kin}\, N$.

*sticking, high friction - Again there are two components along $\hat{n}$ and $\hat{e}$ but now friction is found such that there is no motion along the contact plane also.
Looking at the math, if the contact point is a distance $c$ from the center of mass of the stick the you have the following equations of motion (at the center of mass)
$$\begin{align} m \ddot{x} & = F \cos\theta - N \sin\theta \\
m \ddot{y} & = F \sin \theta + N \cos \theta - m g \\
I \ddot{\theta} & = c N \end{align} $$
Where $I$ is the mass moment of inertia of the stick. It might be approximated as $I = \frac{m}{12} \ell^2$
The condition needed to solve for the contact normal $N$ is
$$ \ddot{y} \cos\theta - \ddot{x} \sin\theta + c \ddot{\theta} = 0 $$
$$ \boxed{ N = \frac{g \cos \theta}{\frac{1}{m} + \frac{c^2}{I} } } $$
The three scenarios have the following conditions


*

*no friction - $F=0$

*low friction - $F  = \mu_{\rm kin}\, N$

*high friction - $F = m g \sin \theta$


Since $N$ and $F$ are now fully defined, use them in the equations of motion to figure out how the stick is going to respond.
