Does the location of the battledome you choose/world matter for the prizes you get? Like are there better prizes (codestones, etc.) in neopia central?

Each arena gives out different prizes, but I'm pretty sure every arena has a chance of dropping codestones. Some challengers award more codestones more frequently than others - Koi Warrior and the S750 Kreludan Defender Robot, for example, drop all 15 codestones and tend to give them out quite frequently.

                  Here you have the prizes in general, and specific place and opponent.

The Premium dome drop Nerks quite frequently, so that makes a difference

Some opponents, as well as domes, have certain rare drops as well.