First, here's a more readable view of one seventh of the circuit between the demultiplexer and the LED array, this time with an additional current-limiting resistor between the transistors:
Speaking of current, the whole circuit draws quite a bit of it, and it may be too much for a single 7805 to handle. Proper heat sink attached to the regulator is at least required, or a different power supply arrangement if that fails to provide enough cooling.
I got some of the LEDs to actually light up, but the addressing appears not to be as straightforward as I had naïvely assumed. More tinkering needed.