Usually when my hands hurt it's because something isn't adjusted right. At the moment all my bikes have drop bars and how the bar is positioned and where the brake levers are on the bar make a huge difference. I've had bars which I could never get comfortable on and had to replace them.

Bigger tires and/or lower pressures can help with the bumps. Coasting over them is fine if you can't go around them but keep both feet on the pedals and get off the saddle a little bit. Keep your arms and legs flexed and use them as shock absorbers.