If I restart the Display Manager, e.g
lightdm, will X be restarted as well ?
It depends on your configuration: you can have X-window Server started by itself and then the Display Manager process or Display Manager could start the X-window server.
I have X server started by kdm in OpenSuse 12.1:
kdm(4655)─┬─Xorg(4671) └─kdm(4698)───startkde(4800)─┬─gpg-agent(4877) ├─kwrapper4(4977) └─ssh-agent(4878)
If you use
pstree you can check by:
pstree -p `ps -H -C Xorg -o ppid --no-header`
ps -H -C Xorg -o ppid --no-header | xargs pstree -p