Wayne, I cut off the beaver tail specifically so I could use it to test the dye, since it was not onscreen anyway. I also cut out the hood, but mainly because I could feel it bunched up against my back and, again, it didn't take away from the screen accuracy in terms of how it looks. Also since I already cut off the tail, the "unseen" authenticity became less important to me. If I had already felt comfortable with the dyeing process, I probably would have left the tail and then also the hood by default. But this was the first time I ever dyed anything, and I didn't want to accidentally ruin the whole coat.
As for the piping, I had assumed it went all the way across, because that just makes more sense from a design perspective, and I uncharacteristically didn't double-check that detail - so I just continued it over across the tab. I think that is the best way if you want it to go all the way (rather than interrupting it at the tab). However, afterward, I did notice that it doesn't go past the tab on the on-screen jacket, so I'm considering chopping it off to be more screen accurate, and would probably recommend that route.
Mine did not shrink at all after 4 dye steps/washings, but I did use a low heat setting when drying it.