I have been using Duplicati to back up Hyper-V guests for many years.
Here is my understanding of how it works (which could be wrong):
Duplicati does not backup the running state of Hyper-V guests, no matter how you configure it. It doesn’t pause/suspend the guests prior to backing them up. It uses the Hyper-V VSS Writer, which in turn uses the Guest Integration Services to request the guests and their applications to make themselves consistent before the shadow copy is made. It’s up to the individual applications on the guest to honor these requests (not everything does).
When doing test restores of these guests, upon first boot they behave as though they were powered off without being shut down. (“The previous system shutdown at XXX was unexpected”). I have never encountered any data consistency problems with any of these test restores, but thankfully they’ve only ever been test restores. I’ve never had to do it for real.
When selecting guests to back up, if you select the guest from the “Hyper-V Machines” part of the UI, then it will include all possible relevant files for the guest. If you choose the individual files from the filesystem itself, you’ll have to keep track of all that yourself. As far as I know, choosing either the entire guest, or all of the guest’s files individually makes no material difference in the backup itself.