Am I multiplexing my LEDs correctly?
LEDs are not powered by a particular voltage level, but by the voltage difference between anode and cathode. That's why you need to drive cathode pins (7, 4, and 2) LOW to switch the LEDs ON and HIGH to switch them off. A truth table could help:

╔═══════╦═════════╦══════════════════════════╦═══════════╗
║ anode ║ cathode ║ voltage anode to cathode ║ led state ║
╠═══════╬═════════╬══════════════════════════╬═══════════╣
║ LOW   ║ LOW     ║ 0V                       ║ OFF       ║
╠═══════╬═════════╬══════════════════════════╬═══════════╣
║ LOW   ║ HIGH    ║ - 3.3V                   ║ OFF       ║
╠═══════╬═════════╬══════════════════════════╬═══════════╣
║ HIGH  ║ LOW     ║ + 3.3V                   ║ ON        ║
╠═══════╬═════════╬══════════════════════════╬═══════════╣
║ HIGH  ║ HIGH    ║ 0V                       ║ OFF       ║
╚═══════╩═════════╩══════════════════════════╩═══════════╝


Your schematic looks safe to me, but be aware that if you decide to illuminate L1, L2 and L3 to the same color simultaneously (e.g drive red line HIGH and pins 7, 4, and 2 LOW), your LEDs will not have uniform brightness. Only one of the cathode pins should be LOW at a time, otherwise you will end up in the situation described here.