For an attended automation triggered from the desktop explicitly, can the automation be run in the background so that the end user can perform another task in parallel?

Yes, the automation should be designed to use commands that use windows drivers instead of mocking keyboard/ mouse actions to allow parallel actions. We also have Picture in Picture (PiP) as an experimental feature in 2020.4.
We have the public StudioX FAQ, so might as well make it easily discoverable as it will likely always cover more StudioX specific questions than a single AREP FAQ