The right way to mix digital signals
Generally speaking, if your mixer has no input or output gain adjustments, the approach you're taking is correct, but conservative.

If the sound sources are uncorrelated, then the power and RMS voltage of the combined signal only goes up with the square root of the number of input signals. For example, if you have 16 inputs, you would only need to divide the sum by 4 to keep the output power about the same as a single input signal. This is a good compromise, assuming you're willing to accept the occasional peak getting clipped (use saturating arithmetic in your adding-up).