Good news: Disabling IOMMU in BIOS allows the Xonar DGX to function (audio is produced through the card's outputs)
Bad news: The Mint 16 seems to need a reminder of what volume level it was at before shutdown/reboot. The starting chime ("login.oga") plays at full volume (OUCH!).
More bad news: In order to get my Xonar DGX working, I have to disable AMD-Virtualization (IOMMU). This has implications for running virtual machines, and is arguably a bug. There are some threads that mention IOMMU at the BIOS level versus at the kernel level, but none go into enough detail to elucidate why this kind of bug might manifest. It's all voodoo to me.
This also reveals that expanded details may be relevant to the problem I experienced:
MOBO: Gigabyte F2A85X-UP4
BIOS version: F4, 03/13/2013
lshw hardware report:http://pastebin.com/GV2uk8EY
Dear forum Gods, Please do not mark this thread as closed.