The DQ6 I believe has more/better features, plus a bigger pricetag. DQ6 has 2x PCI Express X16 slots that support Crossfire, DS3 only has one X16 slot (but 1 extra PCI slot). DQ6 has better onboard audio w/ more features, more SATA connectors (8 on DQ6, 6 on DS3), quad BIOS (DS3 has a dual BIOS), heatpipe cooler on the north/south bridge (probably a little better than the regular heatsinks on the DS3's chips)...
But, both use the same chipset, and are both made to be highly overclockable.
I went with the DS3, because I don't plan on doing Crossfire/SLI for a while longer, and the price was also much lower than the DQ6. Also, I only have 2 SATA harddrives, and my DVD drives are IDE. 6 SATA connections is more than enough for me. On the other hand, the DQ6 has the IDE connection placed better, I really don't like where they put it on the DS3. Anyway, check em out on Gigabyte's website, and compare em for yourself to see which layout/feature list better works for you:
DQ6
DS3
Last I looked, the DQ6 went for around $225, and the DS3 went for around $145 (on Newegg).
And graphics card... Depends on how much you want/how much you can spend. My personal recommendation is the 7950 GT, for the price/performance, I think it's an awesome buy. If that's still too much, I also can recommend the 7900GT, or even lower costing, the 7900GS. If you want the 7900GT, you won't go wrong, it's a great videocard.