I rewatched the end of the epi and, yeah, I was wrong. The Barlow payment happened the week of Marth's murder, not a month later. My below was wrong. I think Ridges used Beck to do the killing, then Ridges killed Beck to tie up loose ends.
[beck is the guy that actually did the killing of Martha (while she was in Denver for cancer treatments), which I think had nothing to do with Barlow or Nighthorse, she was just in the wrong place at the wrong time with Beck. I think Walt figured out it was Beck (in that first month adter her death), which is why he had Henry look into getting Beck killed. When that didn't happen (only beat him up the teeth taken), Barlow and Nighthorse saw a way to frame Walt into killing Beck. The framing backfired though when 'weak detective skills' Falls decided Henry did it. Barlow and Nighthorse didn't want Walt to 'grief himself out of office', they wanted him to be arrested.]
On the Branch/Barlow shooting... It's possible neither died. Barlow only had one shot, so if Branch just wacked his barrel enough to not get killed, both guys are still alive.