Although the 2Wire unit is being used as a DNS server in your situation, the 2Wire's DNS server is not designed to accept dynamic DNS registrations from DHCP clients. This is especially true since your Linksys is being used as an intervening router.

As a result, pinging other computers on the network by hostname is never guaranteed to work unless you were to set up your own DNS server.

In the ping output you showed from the Cojo computer, the IP address being reported there is an IPv6 link-local address. Windows machines will prefer to use IPv6 over IPv4 if there is IPv6 connectivity. The machine pinging itself can use IPv6, so the address reported in the ping output is the IPv6 address. To force the ping command to use IPv4, use:

ping -4 cojo