Ryan,
I'm pretty sure I understand what you're looking for, and will try to give the best suggestions I can. If I'm off in what I'm saying, just reign me in, and let me know how you'd like to do things differently.
1) You can use the "Facebook User Only" User Creation setting of JFBConnect so that when a user clicks the Login button it silently creates them an account in Joomla. I know you may not want an account, but that's the best suggestion for now to not even let them know they created one. There are ways to have them approve your FB application and then get their email address without creating an account, but that's not how JFBConnect is structured at all. Our code always expects that if there's a user you want to get info on, that they have a Joomla account as well.
2) Once the account is created in Joomla, you'll have easy access to their email address (it will be in the User Manager) and many contact forms can auto-populate that field, or link it to the user behind the scenes.. another reason having a Joomla account for the user is good

3) You can have JFBConnect skip user activation, while leaving it enabled for the rest of the site. Generally a good idea since Facebook has already verified the user's email address.
4)You can create your own Login With Facebook button image or text.. "Register for the Sweepstakes Now!" can easily be used. You can simply change the text of the blue-button in the language file.. or you can go hog-wild and use your own image or something craftier. Directions for creating your own Login button can be found on the following page. See the 2nd question:
www.sourcecoast.com/jfbconnect/docs/common-support-questions
Hopefully, some of that gives you some new ideas on what you can do. Of course, feel free to come back with other questions or ideas, and we'll gladly help how we can!
Thanks,
Alex