In certain scenarios with non-linear circuits, such as guitar tube amps, or when adjusted beyond the normal working limits of a circuit, then yeah, gain could impact the tone. It most certainty DOES affect the loudness. Turn the gain up and the final output will also go up. Unless of course, you're already past the limit of the input stage and then it wouldn't increase the loudness, but would increase the distortion.
That's really not relevant in any standard mixer. Different position of gain stages could impact how effects work, and especially may be relevant if you have multiple channels feeding into a single effects unit.
But generally speaking, today's audio gear has a much much wider working dynamic range available and as such the gain settings are very forgiving. Adjusting gain down and turning up the 'volume' fader (I've never personally seen a mixer with a volume adjustment, but that's just me) would have basically offsetting impact on the sound.