I found the solution for the groups issue today: https://unix.stackexchange.com/questions/454593/groups-not-registering-in-x
Uncommenting the lines
auth optional pam_kwallet.so
auth optional pam_kwallet5.so
did the trick.
However, I can’t seem to apply the styling for slick-greeter - the lightdm-settings tool has no affect and changing the theme didn’t work so far. I’d like to remove and reinstall the pocillo theme next, but I’m unsure how to do that (is it an apt package?).
EDIT: I found the solution to that too. Apparently the config file which contained the settings for the greeter-session had been deleted (when purging, previously) and lightdm was thus using the default gtk-greeter.