Message shown: Server authorization directory (daemon/ServeAuthDir) is set to /var/lib/gdm but this doesn't exist.
You could recreate that directory but that would not help much.
The best solution would be to reinstall all packages which create diriectories and files in /var.
The best way to do that is probably to check all installed packages for files in
rpm -qa | while read package; do if rpm -ql "$package" | grep -q /var; then zypper install --force "$package" fi done
You should create
/var/log with these permissions before:
drwxrwxrwt 1 root root 425K 2. Mai 00:53 /var/tmp/ drwxr-xr-x 1 root root 946 1. Mai 23:53 /var/log/
Other directories may be necessary.
Maybe it is necessary to install the packages in the right order. This should be possible by using a single install command with all the package names as arguments so that
zypper uses the dependencies for the right order. I do not know, though, whether dependencies are considered at all during a reinstall.