Well, it looks like a timely restart (perhaps it applied an update, on restart, that fixed something? dunno!) fixed my issues 
I like it that you solved it 
To save Jim from saying it, he advocates rebooting once a week or for troubleshooting 
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Thank you, will keep that in mind for the future.
My faith in “things being in sync” has been vindicated
I had DTTG open on a particular group and saw the update happen in real-time. Very cool.
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Glad to hear it.
btw, “scenario 2” from above still keeps happening (i.e. several files simply show a “Download” link which … if I click on it, nothing happens)
One thing I’ve noticed is that it happens only with “older” databases (which have been around from the iCloud legacy days, and before that, on Dropbox!), but not with the one that I created a few months ago.
Given that, is there some workaround I could try? I’m open to recreating or copying over or anything that might help.
Try “rebuild” the databases in ;DEVONthink?
You indeed can make sur eyour local, primary backups are current and do a File > Rebuild Database on the problematic database.
You could also create a new database and transfer files between the old and new one. However, I would suggest the rebuild first. And I would also as if Date Added is an important bit of metadata for you or not.
Should I do something at DTTG to “refresh things” after I rebuild my database on my Mac ?
Turn it on on IOS device, keep in foreground, and let it sync.