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What province are you in please?
I am sorry for the delay. There is some glitch with the site and my very lengthy answer has disappeared.
I am going to do the answer again but will post by paragraph in case this happens again.
I will let you know when I am done.
You are not correct in terms of your understanding of the law. I Ontario common law spouses do have the same spousal and child support rights and obligations that legally married spouses have. And custody has nothing to do with whether there is a marriage or not.
But common law spouses do not have the same property rights that legally married spouses do in Ontario.
As the house is in your spouse's name she is presumed to be entitled to the full value of the house.
But you can rebut that presumption if you can show that you contributed to the value of the house. So you would have to show that you contributions (financial or through effort) increased the value of the house.
I suggest (urge you in fact) to consult with a family lawyer face to face as your next step. Family law is highly complex and even family lawyers retain their own lawyers to deal with family law matters.
Mediation may work but it best to mediate while be represented by your own lawyer.
So, to get the ball rolling you should consult with a family lawyer and retain the lawyer to send your spouse a letter asking her to retain a lawyer so that the two lawyers can begin to assist you both in resolving the issues.
I am done for now.
Please let me know if you need any further clarification.