Hi. If netclient isn't run in systemmd mode /daemon mode , would Netmaker server push changes to nodes or it would get stale in some time to show warning/error ?

no, you need the daemon running to receive updates

you will see warning/error status in netmaker UI after a period of time

And if its not running means nodes would show warning and error after some time and they won't be able to ping each other..right ?

nodes may be able to still ping each other if no changes to the network or peers have been made