> Does anybody know a way that I can link from my own site to a page of results on the Name Generator that is produced from my own template? Thanks :)

You actually can, yes, provided your templates aren't more than a hundred or so characters long. (Sometimes long URLs get truncated, although that shouldn't happen.)

Try this:

http://www.rinkworks.com/namegen/fnames.cgi?d=checked&c=your_template_goes_here

For example:

http://www.rinkworks.com/namegen/fnames.cgi?d=checked&c=ss(ia)

...for generating names with two syllables, followed by the letters "ia." If you use <|> characters, replace < with %3C, | with %7C, and > with %3E:

http://www.rinkworks.com/namegen/fnames.cgi?d=checked&c=%3Cs%7Css%3E

If triple consonants and/or double vowels are acceptable to you, remove the "d=checked&" portion of the URL.