What's the best way to clean up this rust? (pics)

Evap-O-Rust, you can get it at O'reillys. It works wonders on gears. I had a ring and pinion that were as bad or worse than yours. Soaked the in the stuff and the looked like new. We installed them for a guy and they never made any niose. It's a bit pricey at 24 bucks a gallon, but that's the only down side. Try it out.