Thanks for posting the information, and good to know that the DMX-6Fire-USB works on Linux.
When you have finalised where your page will be hosted it would be good if you could post a brief review and link in this section of the forum: http://forum.audacityteam.org/viewforum.php?f=27
Alsamixer is, as you say, rudimentary, but is very useful as it handles the sound card at “driver level” - that is, it is closely bound to the actual hardware rather than going through multiple layers of API.
Hardware → ALSA (drivers, including Alsamixer) → Gnome/KDE/PulseAudio/Jack
Unfortunately the lack of support by TerraTec is going to make setting up your system a bit tricky and fiddly (but I don’t want to knock TeraTec too much because they are one of the few hardware manufacturers to provide “some” support for Linux, which is to be applauded).
Jack Audio System is likely to be the way to go for getting the best out of this device. It offers low latency, configurable connectivity, and can (now) also work with PulseAudio and programs that are not Jack aware.
Audacity has limited (but definitely usable) support for Jack.
First the limitations:
Audacity does not support “Jack Transport Control” 
Audacity does not (fully) support the “patchbay” in qjackctl.
Enabling the Audacity record monitor before any recording has been done is likely to crash Audacity (a bug).
The Audacity recording/playback meters only have 2 channels (for mono or stereo).
Playback from Audacity is limited to a maximum of 2 channels.
Now the good news:
Audacity works very well with Jack (other than the above limitations).
The Audacity record monitor works correctly (without crashing) after recording has been enabled once.
Audacity can record multiple channels simultaneously into separate tracks.
The first thing to do is to see if you can get basic record/playback functionality in Audacity with Jack.
First you will need to start Jack - do this with Jack Control (qjackctl).
Launch Jack Control and open the “setup”. Ensure that the input and output devices are set to the DMX-6Fire-USB device.
As a first attempt, after checking/setting the input and output devices, try starting the Jack server using the “Start” button on Jack Control.
Getting Jack configured correctly can be the hardest part, but fortunately you don’t need to do this often - once you have it running properly the settings are saved and should work automatically the next time.
When Jack is running correctly Jack Control looks like this:

If error messages appear we will need to fix that before going further.
If Jack runs OK, then launch Audacity and set “Host” to “Jack Audio Connection Kit” in the Device Toolbar. Leave the recording input and playback output set to “System” for now. Try importing a file and playing it. Does it play back OK?
Try recording something plugged into channel 1 of the DMX-6Fire-USB does that record OK?
How are we doing so far?