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

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

No, voltage drop isn't a worry unless it exceeds 5% from the transformer to your outlet then it's not code. Quit obsessing over irrelevant stuff and listen to your electrician. Get him or her to test the voltage on an existing outlet and see what it is. I've had 121 volts on  a 120 volt outlet. It can vary during a 24 hour period. 

Call an electrician and install what code says. Anything here that is spot on with your local requirements is merely coincidence.

As others have mentioned, talk to your electrician.  You don’t need any exotic outlets or special parts for this.  Just ask your electrician to use better than contractor grade outlets.  Make sure you have individual grounds run to each outlet. The ground wires should not be daisy chained.  
If you will be drawing anything near rated capacity on all four circuits you may want to be thinking how you will be keeping your room cool.  If each circuit is loaded with 10 amps of equipment you will be drawing and essentially creating 4800 watts of heat. Which converts to about 16,380 BTU/hour or about 1.36 tons of cooling.

You may want to add up the maximum  current draw of all the equipment you will be using and see what the total is.  Don’t expect to be actually drawing that many amps/watts on a regular basis, only under extreme use conditions will you ever be hitting the max on all equipment.

if all your equipment is placed in a rack or closet you want to be moving a significant amount of air through that space to keep the equipment from overheating.

Forgot to mention:

If you want to go down the rabbit hole that is researching 20 amp receptacles do a search on either Hubbell 5-20R or Bryant 5-20R.  There will be literally hundreds of options.  Check out commercial or industrial grade.  Or for the ultimate check out the hospital grade receptacles.

Secret  tip: many or most of the “audiophile” versions are just rebranded items from the regular electrical parts manufacturers with a commensurate price markup.