How to apply Gauss' Law given charge? 
How I approached this question is by adding the two charges and dividing by epsilon zero. (Gauss' Law)
 Is this the correct way to solve this problem?
 A: Gauss' law states that the net flux of the electric field $\vec E$ through a surface is simply related to the net charge enclosed by that surface:
$$
\oint \vec E\cdot d\vec S= \frac{q_{net}}{\epsilon_0}
$$
Thus only the net charge matters.  
In relation to your specific example, Gauss' law does not states that the flux need be the same through 
all $4$ sides of your pyramid - the flux through individual surfaces depends on the location of the charges - but is a statement about the total flux through the $4$ sides.  
To see this net flux does not depend on the location of the charges, imagine an analogy with a 60W lightbulb surrounded by a cubical lampshade.  If you move the lightbulb around, the total  amount of light through the lampshade will not change - that's determined by the 60W lightbulb only.  Of course if you move the lightbulb close to one side, more light will escape through that side but less light would escape through the other side, without changing the total amount of light through all sides.  If you want more light through your shade, you need to increase the 60W to a 120W bulb, which is the equivalent of increasing the charge enclosed in your Gaussian surface.
