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1. Yes, she can ask him to leave. If he pays any of the mortgage, he would be considered a renter and would be entitled to 30 days notice to vacate.
2. They both have rights to any property they purchased, and also, each one has full right to the money in the bank account. My suggestion is for her to get out what is hers and put it into an individual account.
3. No, he doesn't have any right to the house or equity in it. He doesn't own the property, thus he's got no entitlement to anything.