In the current search engine-speak, "sponsored" is synonymous with "paid." I suppose some marketing study showed it to have less of a negative connotation than other words that might be used: "paid listing," "advertising," "rankings influenced by large cash payments," etc.
If you submit via their paid inclusion service ONLY the page paid for is spidered, no links are followed.
If the spider finds the site on its own or you submit via the free add url page on alltheweb then the spider should follow the links it finds. It depends on how search engine friendly the link structure is.
"Sponsored links" on Ask Jeeves and Teoma are provided by Google. Is it possible that you are listed with their AdWords product? If so, that would explain why you are appearing on Ask Jeeves and Teoma as a "Sponsored Link." If you submitted a url to the Ask Jeeves Site Submit program, and don't find it within the "my search results" section of Ask Jeeves, please don't hesitate to contact INeedHits at email@example.com or by calling them toll free: 866 873 5834.