Check the bindings on the DHCP server and make sure DHCP is bound to the correct networks. There appear to be issues with dhcp communicating with dns according to the logs.
Check the lease durations to make sure they aren't too short, or too long "The log was temporarily paused due to low disk space" That log entry didn't scream trouble to you??? lets see a ipconfig /all from the server and a workstation with a working lease You have an interesting mess. 8GB RAM From the client 192.168.0.1 is the GW to the server because it is the router too. :/ ) From the server the GW is the NIC other port which gets the internet.
Have you tried stopping and restarting the DHCP service on the server?
Open the DHCP mmc and right click the server, and select Reconcile all scopes.
I tried restarting the DHCP server the Full server sevral times but no success.
lack of disk space is the best way of permanently lose your server with no recovery except to wipe and restore from backup. I would still run chkdsk /f on each drive to check for file/disk issues. C:\ has the file system and it is mirrored Yes 192.168.0.1 is the DNS too and there is No error in the log Yes it is configured to update dns, I tried with switching off but didn't help.
Lack of disk space causes problems with the paging file, promotes severe fragmentation and generally causes windows to slowly die. For me it is a nightmare : D the left is the server the right is the client on the server the primary dns entry should be itself at 192.168.0.1 not google.