Does the sound change pitch with engine speed?

Not really.

Does the sound change pitch with road speed?

Yes, but not consistantly one way or the other. It's usually loudest at around 34-45 mph on smooth road. At higher speed, it seems to be less loud. I'm not sure if it's the present or lack of road noise that makes it louder or quieter...

Riding the brake has little effect.

Oh yes, the "groaning and moaning" noise is usually accompanied by mild vibration of steering wheel and gas peddle, of similar frequency.