Ralph,
Yikes, that's a lot of errors.
Most of the errors do appear to be related to network communications. Most of the captures/verifies/repairs that fail predomenatnly report POSIX error 60 or 6.
Error 60 is an "operation timed out" error, usually associated with a network communications socket or device channel. Error 6 is a "no such device" error; in this context it usually means the volume/drive being addressed is no longer connected.
A lot of your failures follow the pattern of getting "timed out" errors, later followed by "no such device" errors. I suspect you're having network communications or remote storage device problems that initial stop responding to requests, and later appear to go off line. You can see this in events such like that starting on May-12 where the first capture starts but dies with an error 60. Subsequent actions then fail because they can't access any archive files (error 6).
I also see that your archive's volume tends to get mounted and unmounted a lot. At first I thought this might be an indication of a problem, but the timing wasn't quite right. Instead, it seems to be by design; you apparently mount the volume and then manually start a capture action. (Just FYI: if the volume can be automatically mounted, QRecall will mount, and unmount, the volume for you.)
I did find one really suspicious sequence of events that I think lead to all of the problems on May-9. The volume was mounted at 12:23 and the capture action was manually run a few seconds later. The capture action ran until 13:17, at which time it encounted network timeout errors (60) that prevented it from finishing. But before that, it appears that the system was put to sleep:
2016-05-09 13:16:30.695 -0700 Power Management will go to sleep
2016-05-09 13:23:29.285 -0700 Power Management did power on
(Please note that there's another possible problem here. Starting somewhere around OS X 10.10, the kernel is letting background processes run in the so-called "power nap" mode, where the system is mostly asleep but some background processes are still running. Unfortunately, QRecall seems to be one of the processes that it's allowing to run, but not enough of the rest of the system is awake to function correctly and errors ensue.)
It's been my experience that network sockets don't like to be put to sleep, and can take quite awhile to recover when the system wakes up again, which is probably the source of that particular failure.
Without anything definitive to go on, I'd recommend trying to isolate the pieces one at a time and see if you can find some improvement.
First, is it possible to eliminate the network and server for a trial and connect the archive drive directly to the system, just to make sure it's not the drive or something else?
Then try to replace pieces one at a time to see if that makes any difference. Try a different network connection. If you're using WiFi, try to hook up a hard-wired ethernet cable. If you're using ethernet, see if you can use IP-over-Firewire or something. Can you move the archive drive to a different server?
I know I'm not being terrible helpful, but I hope it gives you some ideas to try.