Sorry guys for not responding to your questions. I somehow didn't get the notifications about the discussion here. Also: I didn't really look further into this, because of a simple workaround I had to implement: remove all snap apps and install them from the repos.

Now I did recently upgrade all our computers to Ubuntu 18.10 and the issue still persists, so I'm using the same "workaround" as well, but I will keep one computer as is (with snap apps), so if someone would like to know how things are working (or not working) here, I will be happy to provide you with more details.

Current setup:
snap    2.36.2
snapd   2.36.2
series  16
ubuntu  18.10
kernel  4.18.0-11-generic

/home is mounted via fstab at boot time like this:
10.0.1.98:/home /home nfs4 rw,hard,bg,sec=krb5 0 0

Running a snap app will not do anything for the user, but will report errors like these in the journal:

gnome-calculator      3.30.1                  260   stable/…  canonical✓        -

dec 13 15:50:36 jerusalem dbus-daemon[2172]: [session uid=10000 pid=2172] Activating via systemd: service name='org.freedesktop.portal.Documents' unit='xdg-document-portal.service' requested by ':1.124' (uid=10000 pid=4948 comm="/snap/bin/gnome-calculator " label="unconfined")
dec 13 15:50:36 jerusalem systemd[2095]: Starting flatpak document portal service...
dec 13 15:50:36 jerusalem xdg-document-portal[4958]: error: Failed to load db: Chyba pri otváraní súboru /home/michalmaria/.local/share/flatpak/db/documents: Adresár alebo súbor neexistuje
dec 13 15:50:36 jerusalem systemd[2095]: xdg-document-portal.service: Main process exited, code=exited, status=2/INVALIDARGUMENT
dec 13 15:50:36 jerusalem systemd[2095]: xdg-document-portal.service: Failed with result 'exit-code'.
dec 13 15:50:36 jerusalem systemd[2095]: Failed to start flatpak document portal service.
dec 13 15:52:36 jerusalem dbus-daemon[2172]: [session uid=10000 pid=2172] Failed to activate service 'org.freedesktop.portal.Documents': timed out (service_start_timeout=120000ms)
dec 13 15:52:36 jerusalem gnome-calculator_gnome-calculator.desktop[2311]: cmd_run.go:832: WARNING: cannot start document portal: Failed to activate service 'org.freedesktop.portal.Documents': timed out (service_start_timeout=120000ms)
dec 13 15:52:36 jerusalem gnome-calculator_gnome-calculator.desktop[2311]: cannot change current working directory to the original directory: Permission denied

Another example:
spotify               1.0.94.262.g3d5c231c-9  28    stable    spotify✓          -
dec 13 15:55:18 jerusalem dbus-daemon[2172]: [session uid=10000 pid=2172] Activating via systemd: service name='org.freedesktop.portal.Documents' unit='xdg-document-portal.service' requested by ':1.138' (uid=10000 pid=5160 comm="/snap/bin/spotify " label="unconfined")
dec 13 15:55:18 jerusalem systemd[2095]: Starting flatpak document portal service...
dec 13 15:55:18 jerusalem xdg-document-portal[5172]: error: Failed to load db: Chyba pri otváraní súboru /home/michalmaria/.local/share/flatpak/db/documents: Adresár alebo súbor neexistuje
dec 13 15:55:18 jerusalem systemd[2095]: xdg-document-portal.service: Main process exited, code=exited, status=2/INVALIDARGUMENT
dec 13 15:55:18 jerusalem systemd[2095]: xdg-document-portal.service: Failed with result 'exit-code'.
dec 13 15:55:18 jerusalem systemd[2095]: Failed to start flatpak document portal service.
dec 13 15:55:18 jerusalem PackageKit[1825]: resolve transaction /593_caaeadbe from uid 10000 finished with success after 391ms
dec 13 15:55:18 jerusalem gnome-software[2834]: State change on system/snap/Obchod Snap Store/desktop/io.snapcraft.spotify-pOBIoZ2LrCB3rDohMxoYGnbN14EHOgD7/* from available to installed is not OK
dec 13 15:57:18 jerusalem spotify_spotify.desktop[2311]: cmd_run.go:832: WARNING: cannot start document portal: Failed to activate service 'org.freedesktop.portal.Documents': timed out (service_start_timeout=120000ms)
dec 13 15:57:18 jerusalem spotify_spotify.desktop[2311]: cannot change current working directory to the original directory: Permission denied