Ok, here I am now, a few months after the original post... and here's the situation.
I've found that yes, I can run two things simultaneously from the Chaintech AV-710. However, for whatever reason, i have to set the audio deck to 2 Channel - NOT 2 Channel High Sample Rate. If I set it to the later, only one output is available (the one with the black hole, closest to the infrared port). When I set it to 2 Channel, the green hole outputs sound. I suppose there's no way around this, so a splitter may be an option, correct? However, would this have any advantage, in terms of convenience, over simply swapping between headphones and speakers via their respective cords?
For reference, my amp, etc is in the sig, and my speakers are Altec Lansing MX5021.