"passive" XLR adapter and mixer. Passive because it has no amplifiers, is a "mixer" because there are separate audio gain controls for the left and right channels and is an "XLR adapter" because it provides the correct signal and power to camcorders using a 1/8" (3.5mm) stereo audio input. But you already know this.
The Sony ECM-44 lavalier mics require a power source. When used wireless, the Sony WRT-805 transmitter provides power to the mic. When used wired, power is supplied by a AA battery in the XLR connector. The ECM44 does not use phantom power like that supplied by many mixers. The DXA-4 does not have phantom power capabilities - but the DXA6 does (this is irrelevant when using the ECM-44). You probably already know all this, too. I just want to be sure we are all on the same page.
The camcorder does not know the difference between connecting multiple mics or anything else. It knows only that there is a 1/8" (3.5mm) audio input plugged in.
* on the DXA-4: between the two XLR inputs, there is a MIC/LINE switch (one for the left channel and one for the right channel) and both are switched to MIC.
* on the DXA-4: the audio gain knobs are both set at about halfway between 0 (low) and 10 (high).
* on the Canon HF G10: manual audio is the selected option setting. See page 67 in the manual
This single manual audio gain control in the camcorder controls the left and right channels at the same time so there is no separate individual channel control. For this testing, the gain should be about 1/2 way between low and high. And the audio levels should be visible on the camcorder viewfinder.
Plug in a known working ECM-44 to the DXA-4 - and plug the DXA-4 into the HF-G10. The audio level meters on the camcorder should show audio. Move the mic to the other channel. Use headphones to monitor the audio. See page 71 for headphone use.
Does the audio follow the mic when the mic is moved from one channel to the other? If yes, unplug that mic and do the same with the other mic. The results should be the same. Now plug in both mics. What happens?
Please clarify "record audio tracks without video". When the camcorder is recording, it is recording video and audio. Your video editor can strip out the audio if needed. As far as I know, the HF G10 is not an "audio-only" recorder.
Yes, it is possible to have the external mics and the internal camcorder stereo mics recording at the same time. See page 70, "Audio mix".
How are we doing so far?