I'm Lucy, and I'd be happy to answer your questions today. I'm very sorry to hear that this happened.
No, asking you to initial a document on her behalf would not nullify the mortgage. Even if it did, I'm sorry to say, that wouldn't really help your case. The bank could call upon her to repay the mortgage, and she could refinance. But it's not fraud if you initialed something on her behalf, with her consent.
Your ex-girlfriend is unfortunately correct. A person who lives with a partner who does NOT get married has no interest in the property they live in together if his or her name is ***** ***** the title or mortgage. The only exception is if you have a written agreement where you can prove that she agreed to repay you for a portion of your contributions in the event of a breakup. Otherwise, the money you paid each month is treated as rent. You were essentially paying her to live there instead of a landlord. Any additional amounts you paid is treated as a gift. Money tends to flow back and forth during a relationship. A judge will not go back and look at who paid for what after a breakup for unmarried couples. The two of you made a choice not to get married, and not to put your name on the house, and that's unfortunately not a decision a court of law can undo for you.
The only way you'd be able to collect any money here, other than producing a written agreement where she says you're entitled to half the house, is if you could establish that you gave your ex-girlfriend a loan for any of that money, like to do specific repairs or improvements or something like that. Otherwise, the house belongs to her, and she has no legal obligation to pay you anything.
I apologize that this was probably not the Answer you were hoping to receive. However, it would be unfair to you and unprofessional of me were I to provide you with anything less than truthful and honest information. I hope you understand.