My router handles dhcp, not the cable modem...From Netgear...By default, the router acts as a DHCP server. The router assigns IP, DNS server, and default gateway addresses to all computers connected to the LAN.
Are you saying when the router isn't connected the Netgear cable modem will provide dhcp services?
The modem operates at Layer-2 of the networking protocols, aka the MAC Layer. Your public IP address will be what Cox assigns to your modem. When your modem receives data addressed to it, it'll pass it via MAC address. No IP addresses involved.