Replace panels before or after dipping?

Dip first. Any metal you cut away will weaken the structural integrity of the unibody. I have watched a BUNCH of videos of cars being dipped on YouTube (it's actually quite interesting), and they have never been cut up before dipping. I have also seen a bunch of car shows where the cars were sent out to be dipped, and they also were not cut up before.
Call the place you are thinking about using and see what they think.
Also take a look at some of the YouTube videos. They are interesting and informative.