Reverting to Logitech SetPoint 6.32.7 seems to have resolved the media keys issue. They now work regardless if MusicBee is the active window or not.
As for the other issue, it seems that it is playing the tracks in order of their date for the affected albums, and then by track. Therefore making all of the dates the same "fixes" the issue. I still would like a solution that preserves each tracks release year though. Some (I'm not sure about all) of the affected albums exhibit another odd behavior. In "Artwork" view mode, if I double click on, say, the first track of an album, but that track isn't the first chronologically, it actually plays the first chronological track rather than the one I clicked on. This behavior continues to the second track, which actually plays the second chronological track, etc. If two tracks have the same year, the lower track number is played first.