not sure...
what would the proper launch and quit protocol be in terms of Bitcoin Core and Armory?
So, that setting I had you uncheck? That means you need to manage Bitcoin Core yourself, rather than just launching Armory and having it sort things out. One extra step, but I've found it a lot more robust and is my go to when things aren't working automatically for people - but
you need Bitcoin Core to be running for Armory to go online.
So, usual process for me:
Start Bitcoin Core from the shortcut (which loads up the Graphical User Interface [GUI]). Wait for the splash screen to finish verifying blocks.
Start Armory.
Wait for everything to catch up and then, usually, Armory will pop up and say everything is in sync and online.
To shutdown:
Quit Armory. Optionally check if ArmoryDB has closed too (It doesn't seem to shut down if it is busy)
Quit Bitcoin Core.
Does that help?