Some airlines have permanent invitations posted in "Partners" sections of their sites. Examples are BA, Air France, BMI, you can check the websites. Your reward choices will be limited to Hilton, PC and the airline, or just BA if you choose BA.
If you are subscribed to Blockbuster for some time, you maj get a Blockbuster invitation, which will allow you to redeem to any airline partner, except BA.
I suppose the number of surveys received depends on your sponsor.