1 - Only Foreground Triggers work in a user's regular account - because the background service only runs as root and uses /root/.TRIGGERcmdData.
2 - Root can support both Foreground and Background Triggers - because you can install the background service (see #1), and because you can login as root and run the agent in foreground mode.
3 - But running Triggers as Root will mask/override all the Triggers in your regular account - I wouldn't say mask/override, it's just separate as you covered in #4. Running the agent as root will create another separate computer entry in your account associated with /root/.TRIGGERcmdData. That computer will be named the same initially, but you could rename it.
4 - The exception to (3) is if you can instantiate two "Computers" -- one pointing to Root for the "Background", and a separate "Computer" pointing to your normal Linux account to support the "Foreground" - correct, although as you pointed out in #2, you could also run the foreground agent as root.
You create a separate computer just by running the agent as the other user. If the ~/.TRIGGERcmdData directory doesn't exist, the agent will prompt you for a token, create a computer record in your account, and store the ID for that computer in the computerid.cfg file.
Different Linux accounts could setup different computer records, or you could give everyone sudo access to be able to edit /root/.TRIGGERcmdData/commands.json to run background command via the background agent.
The agent stores the .TRIGGERcmdData directory in the home folder of the user running the agent. The files in that directory are the key point of connection to the mother ship.
Good questions @Joe.