Please understand -- If you set your 655 to use mynetwork.org as your suffix, then the DNS server at mynetwork.org would need to be notified of the ip address and host name assigned to that computer. Since the 655's DHCP facility can't update the DNS records on mynetwork.org's server, then there is no possibility that would ever work. Hence the need for a local DNS server.
I hate to say it but you need a clearer understanding of what DNS does, how to use it, and where & what it applies to. I'm not being mean, only realistic. Someone who understood DNS wouldn't ask a question like that.
Regarding your other points:
A local domain name is completely optional- that's actually a DHCP option (is forwarded to a client when as its connection-specific dns suffix). If you leave it blank, the 655 will assign a suffix provided by your ISP, if any.
It's as optional as the "Host Name" field under Setup-> Manual Internet -> Internet Connection Type (DHCP).
Please understand that Windows has mechanisms to detect other computers on the network without a DNS server present on your local subnet. 99.9% of home systems behind a router work without any issue in this manner - all the way down to Windows 3.11.
I can't speak for other vendors products-- but if you look up dd-wrt (a replacement firmware for Linksys and other similar routers), as advanced as it is over stock Linksys firmware even it doesn't have a "proper" DNS server, it only maintains a cache with local and isp dns servers. Meaning, if you don't have a DNS server on your private subnet, the router won't know the DNS names of your local systems. (
link)
If you want to know what systems your Windows computer is aware of via NetBIOS, drop to a command line and type "nbtstat -r" and it will show you the list of systems and workgroups for which it knows on your network.