BR
Yes, I agree it is best practice to shut down applications before syncing in both direction and sometimes when backing up their files. (EDIT: for the sake of clarity I am only talking about an applications own product files, such as Calibre's libraries, so nothing to do with the operating system, for example).
But is not a problem, as far as I can see, with Calibre and FFS when the purpose is backing up as if a file is in use at the time of syncing it FFS gives the opportunity to leave it out, or to retry after one shuts the offending application down or waits for it to finish (as one could in Calibre's case as it does not permanently lock library files throughout a session, unlike, say, as MSOutlook does with its .pst files).
I, personally, would not rely on Calibre maintaining database integrity if syncing libraries in both directions (so not a simple backup) over multiple PCs if the synced libraries are to be edited on multiple computers, and such multiple PC editing is the only reason that comes to my mind for wanting to sync. in both directions between PCs in the first place. So, for myself, the question of shutting Calibre down while doing such a sync is not something I have ever had to contemplate
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Yes, if one has all the libraries under a parent folder and then syncs that parent folder, FFSs own tracking files are invisible to Calibre (and hence to its Library Maintenance tool).
For myself, while I do have all the libraries under single a parent folder I have each library folder itself stacked separately within in a single FFS job, the decision making behind that being driven only by my desire that if something gets screwed up during a sync it will be likely limited to one library, not all. Entirely a personal decision and perhaps ultra conservative.