I'm of the opinion that when building a gaming PC, if you plan to use any decent resolution (1920x1200 for example) you should invest in a higher end graphics card.
Deciding on a $500 budget that includes your gfx card, headphones and a DAC/amp means that some serious compromises are going to have to be made.
There is a TON of undeserved bashing of PC on-board audio in my opinion. I think this is probably a perpetuation of the fact that many years ago on-board audio was very weak. But enthusiast level gaming motherboards do actually have decent sound chips. Mainstream digital audio as a format has not changed much since the earliest days, but the audio processing chips certainly have and even the less expensive processors (DAC chips) sound really good compared to what we saw 10-15 years ago.
I stand by my earlier comment and say go ahead and get that GTX780 and use your on-board audio. Trust me, those smooth frame rates and high-res textures make a gaming rig one you'll thoroughly enjoy and later on you can invest in the audio part,