All of those channels that are in HD are 720p. I believe DirecTV only has FXM in SD. I'm not sure about Nat Geo Mundo and Fox Life.
If your local Fox affiliate broadcasts in HD, then their OTA (not satellite) feed should have the absolute best PQ since that OTA has the least amount of compression. As for DirecTV HD feeds of the mentioned channels, I believe they would all be more or less the same in PQ. Maybe someone with more experience than me in DirecTV's compression can further assist.
EDIT: 720p is usually better at producing high speed images than 1080i, which could be the issue you mentioned with your local PBS station (PBS is 1080i). This is the reason that Fox and ABC networks chose to go with 720p in the beginning of HD - at that time 720p could produce better images for sports.
I'll be watching the show as well and would want the best PQ as well. Personally, I wish there was a 1080p feed of the series but I imagine we'll have to wait for it to be release on BluRay before we get that option. I plan to watch it on National Geographic Channel, and have the DVR record it on one of the other channels as a backup, probably FX.
As for why the show is being aired on so many channels simultaneously, I can't imagine what the idea is behind that. Seems kind of silly to me.
Edited by WB4CS, 07 March 2014 - 11:51 AM.