Because you have a child, you have a duty to provide a home for dependent children and hence, there is every likelihood that she could remain in the property until the child reaches 18. She has a financial claim on the property if she has contributed towards it, mortgage or repairs or improvements, during the time of the relationship but she has lived there.
No, you would be liable to pay child maintenance and she would be liable to pay you an occupational rent in respect of living in the property.
If they move out, you are only liable to pay child maintenance and any rental agreement comes to, is entirely up to her.
I am pleased to have helped. Best wishes