2010 Mac Pro suddenly getting stuck booting
Not really an answer, but what I'd try next to try to reduce the problem cause further...
This can be tweaked if we get closer to a solution.


Remove all other drives, PCI cards &amp; any external USB devices.
Reduce the RAM to one stick [or one matched pair]
Put the original GPU back, if it's been swapped out [&amp; you still have it]
Boot to Safe Mode [Shift key at the chimes]
If you can get to Safe Mode, make a new admin account &amp; try logging into that.
If possible, set up a known-good boot drive on another Mac Pro &amp; swap it in [This is possible between 3,1 up to 5,1 [I haven't tested other models], a boot drive made on one machine can be swapped directly to another, up to El Capitan but not Sierra]
Test if it can boot from Time Machine directly &amp; if a restore from there works.
Test if it will boot from El Capitan or Yosemite rather than Sierra. I had some odd issues with one El Cap security update which would leave me staring at a white screen &amp; going back to a point before it was my only solution.
Security update 2016-003 10.11.6 issues - which has no real conclusion, just a whole lot of working method. Might be interesting to read if you're bored ;-)