How to enable the DEBUG privilege for a user in Active Directory

This should happen automatically (for windows machines on your domain as long as 'register this connection's address in dns' is checked under dns tab). If you change the IP manually or if it receives a new IP via DHCP this should be automatic. You can delete them all from DNS and they will eventually all show back up.

As mentioned, ipconfig /registerdns will force registration, but your comment that it doesn't work "for a dropped and died computer" is confusing. If the machine is off the network or powered off then there is no way for it to register its address. Please provide more information if possible.

You can manage the network configuration on a system without having local Administrator rights if you are a member of the "Network Configuration Operators" local group; so you will need to add your domain users to this local group on the machines where they need this specific right.

Whatever is better to do this via group policy or manually depends on if you want those users to have this right on every computer or only on their own one.