1. my assumption is that 'audio mixing' refers to the process of turning our buffer's data into analog audio output on the various channels[...]
is the process of combining several audio source. In handmade hero we mix to a single track with two channels (left and right). For example, in a game, it's the process of combining the music and sound effects into a single track. It doesn't include the digital to analog conversion which
is done by a digital to analog converter
circuit or chip.
For your second question: keep in mind that Direct Sound is emulated since Windows Vista so even if you choose hardware mixing it would not change anything in our case.
But even before that it wouldn't change in the case of Handmade Hero. Casey always does the mixing himself, meaning he only have 1 Direct Sound buffer, and he combines the different sounds he want in software and write them in that buffer. So Direct Sound only has one buffer and no need to mix anything.
To my knowledge, the Direct Sound software and hardware mixer mix several Direct Sound buffer (you would need to create several of them with CreateSoundBuffer) and possibly (I'm not sure) apply effects on them (e.g. reverb). In that case the hardware mixer would possibly have been faster. In this case, you would create a buffer for the music, and several buffer for sound effects. You would fill them up directly with the complete track or sounds (or maybe in big chunks but you would not be chasing the cursors) and start playing them. The Direct Sound mixer would combine them itself.