A birth certificate is required. Wait for the RFE to see if anything else is missing so you can send it all at once. Keep a copy of everything you submit and use mail service that allows for tracking. Good luck.
First, have a consultation with an immigration attorney in your area. The attorney will review the documents and advise you on the best course of action. There is a difference between leaving voluntarily and being granted voluntary departure by an immigration judge. If you leave voluntarily while in proceedings, you are self deporting.
It should not be a problem, as long as she did claim to be a U.S. Citizen. Nonetheless, you should talk about this and any other questions with an immigration attorney in your area before you file anything. Good luck.
Canadians do not qualify for DV-Lottery 2014, unless they have a 2nd nationality/citizenship from eligible country.