Do you think that some company like GoDaddy or Host Gator would trade a domain name or free hosting in exchange for me putting ads to their site on my site?
You would probably find loads of people on here that would do that, but a better way may just be to sell the ads on here, and then use the money from that to buy the domain, that way you make more money.
To answer your specific question; No, the companies are too big to really be able to involve themselves in a "domain by domain" case. But Nuke13's suggestion is probably the best idea, and you can honestly get more for your time and money as Godaddy and Hostgator probably won't be willing to pay much, but if your site is targeted, has any decent traffic, and the like -- you might be able to get a nice penny for link or ad sales =).