Copy to tape stage hanging with status "active" but "show activity" greyed

After a period of relative stability, my BE2012SP4 installation starts acting up again. I came back from the weekend to find that various "old friends" are back:

• one of my AVVI jobs claiming one of the VMs it should back up had spontaneously developed a "disk that doesn't support GRT" (V-79-57344-38725)
• another one claiming that the RAWS on another VM had equally spontaneously mutated into an "incorrect version" (V-79-57344-33899)
• the same job complaining that on a third VM, both the Symantec and VMware VSS providers were installed (V-79-8192-38331)
• and  on a fourth VM, that it couldn't access the metadata for GRT (V-79-57344-38726)

From past experience, all these problems will magically disappear on the next round, just leaving a stale impression of being not quite predictable.

Much more annoying, however, is the fact that once again, none of the copy to tape stages of the weekend's full backup have run, because a previous job was still occupying the tape library. The job in question is listed on the all servers - jobs page as "Status: Active", "Job Status: Active" (just that, no elapsed time or number of bytes done), but "Schedule: Completed", and the "Show job activity" position in its context menu greyed out. (See attached screenshot.) The "Cancel" option was available but did not have any noticeable effect, not even after restarting the Backup Exec GUI.

A usual, the next step was to restart the Backup Exec services. As usual, the Backup Exec Job Engine service didn't shut down gracefully and I had to kill it via Task Manager. After restarting the services, the hanging job was indeed canceled, but the waiting tape jobs still hung with "Job Status: Ready; no Backup Exec servers available". (See second screenshot.) Before the service restart that was: "Ready; no inactive devices available". Half an hour later, the first one finally started and the others went back to: "Ready; no inactive devices available".

Can anyone explain what happened here and how I can avoid that in the future?