Sorry for the delay in replying. I was traveling for work.
OK. So I ran Windows Update and it said "3 important updates, 2 selected". I figured I wouldn't mess with that and tried to install the updates it had selected. I got a message that one had failed. (My apologies, I didn't record the error number)
Oddly, when I tried again, it said there were 7 important updates. 1 failed on that run. Error number 8007312.
Ran WU again. 7 updates, 7 selected. I succeeded, 6 failed (Error numbers: 80073721, 8007371B, 8000FFF)
Restarted and ran WU again. 1 important update. 1 failed (Error number: 80073712). When I went to look at the list it showed SIX updates were actually available: Security Updates -KB2992611, KB3003743, KB3011780, and KB302366 (the one that was selected) and Update for Windows - KB3008627, KB3013410)
Shut down and Restarted.
Ran WU. 1 important update (KB3023261). Failed (Error number: 80073712) "Review Important Updates" showed same six as above.
Restarted and it showed "Installing 1(2,3,4,5) of 5 updates" message.
Ran WU again. Same six updates available. Six failed. (Error numbers: 80073712, 8007371B, 8000FFF).