problem with LED Matrix circuit
The RED LED should not be glowing. Maybe something is wired wrong or the 2N7000 is damaged, but there should not be more than nA of current flowing with 10mV on the gate, which will not produce a visible amount of light in normal room lighting, even with the most efficient red LED. Try replacing Q2 after checking the wiring. 

Of course if you are leaving some types of multimeter connected where you are measuring the 0.35V then there could be enough current through the meter to provide a faint light from the LED. 

There is no problem in reverse biasing LEDs up to the rated voltage (usually 5V), and in fact far beyond that typically (but there are no guarantees so you shouldn't do that).