Dedicated 20a lines, 125 feet distance, which gauge wire to run?

First thing you should do is figure the current draw for your equipment. 
A little overkill is ok but why get carried away?

In a previous job I rewired our ISP buildings with #10 stranded for all 110 VAC circuits, always with a separate neutral (no sharing with other circuits). This was in case we ever needed to change an equipment rack from 110 to 220 (or possibly 208). And yes, stranded handles more current and dissipates heat better.  

Also a common mistake in sub-panels, whether 110 or 220 is bonding the neutral to ground. The is the rule in a main panel but a no-no for sub-panels. 

Where I live it’s illegal to run the four separate lines, so I’d run a 220 line to a sub panel and then run my separate lines from there. It’s also much easier and safer that way.

All the best

JD

125 feet is quite a long run and you should consider using a subpanel. Even with #8 wires you will lose 3 volts at 15-amp draw. A subpanel with minimum #4 wire will limit voltage drop to a little over one volt at total current draw of 15 amps. From there go with 10/3 romex to the outlets. The 10/3 comes pretwisted and that helps eliminate common mode noise. Never run 10/2 parallel romex, that acts like an RFI antenna.

Does more voltage drop also hurt sound quality? Or does it just affect how much total power my system can draw at a given time? 

''Does more voltage drop also hurt sound quality?''
Yes.