As mentioned in the Gi specs, it is also an audio/MIDI interface. There should be no reason to use the integrated soundchip.
Read the manual carefully. Buying 1000 dollar hardware you should study carefully how to use it. I'm not sure what it exactly means that the USB-port supports both audio and MIDI. It is quite possible that you're right in assuming it can't transmit audio from the Gi itself, but that would be odd, because then you'd need another soundcard for recording the Gi audio.
About the way you're doing things now:
ASIO drivers are not supported by integrated soundchips, and MME is the poorest driver there is. You could try downloading free ASIO4ALL driver, which works for many. It's not ASIO, but a WDM wrapper.
What kind of a cable do you use connecting Gi to PC? If you want to record in stereo, you must use both left AND right outputs of the Gi, and use an adapter for getting them in one stereo plug to put into PC, or use the headphone output, which is stereo. There can not be a stereo sound without two sound sources. I assume the audio input of the PC is stereo, it most likely is.
Getting audio only on one side of the stereo track is simply due to the fact that you input only mono/left signal. This is the same in all real DAWs, it's nothing SONAR specific. I don't quite understand how Audacity could do it any other way. If it only doubled the mono signal on two tracks, that is NOT real stereo.
Also, it's no use recording in stereo, if the sound itself is not stereo. As a thumb rule, vocals, guitars, basses and similar mono sources are recorded in mono.