How do I calculate the gain of an op-amp summing amplifier?
Gain is set for each \$ V_{in} \$. 

\$ A_{Vin1} = -\dfrac{R_3}{R_1}\$. 

\$ A_{Vin2} = -\dfrac{R_3}{R_2}\$. 

and so on, \$ A_{Vin N} = -\dfrac{R_3}{R_N}\$. 

If you were to analyze this circuit using superposition, you would find that for each input signal, it is just an inverting amplifier. The signals are amplified or attenuated individually. Therefore, gain is only a meaningful quantity with respect to each individual signal. Since gains are all independent of each other, summing them does not yield a meaningful result.