On psr3000 or stand-alone processors, can you control how much your own voice stands out in the harmony mix? Can it be controlled easily for each song or do you have to push a lot of buttons?

EZ as pie. Write different settings into registrations .... even link them to midi files within the registration.
The outboard stuff usually has dedicated knobs for balance ans well as presets.
Visit www.boydsongs.com
Go to the songs page, and listen to Nightengale. It's LOADED with harmonies.