Tassie Devil,
Have you had a look on the Belkin web site? I know someone who had a similar issue and I think they got a download from the Belkin site to encrypt their communications. Once you have a channel between your 2 computers that is encrypted with a key that your neighbour doesn't have, it ought to block outside communications.
If it still tries to communicate with your neighbour's network after looking at that, then try changing your IP address range and/or network mask.
As for the second question, I'm not really sure but, from personal experience of home LAN issues, I'd try disabling any firewall software (on either end) for a minute or two (whilst being disconnected from the Internet of course) and see whether that improves things.
Let us know how you get on
