I see hosts sometimes have this
'Our Network Solutions Nic Handle is BAxxxx.'

Is that some special code to communicate with Network Solutions to point the domain name (registerd thru NS) to the host's DNS numbers?

No, basically it is just a set of contact info. Instead of re-typing name, co name, address, phone, etc., you can just enter in the Nic Handle.

You either use one you have or type in an address and one will be automatically assigned to you (that you can use again).

You can get one here:

http://networksolutions.com/cgi-bin/makechanges/itts/handle

Best Regards,
Matt Lightner
mlightner@site5.com