Do I have a common law marriage?
No, there is no such concept as a common law marriage. If you are not legally married, then regardless of the lengthy of your relationship, you have no right to a financial settlement via the family court.
Child maintenance is always payable by the non-resident parent, even if you were not married to each other, and you can calculate the amount due here: www.gov.uk/calculate-child-maintenance
There is also provision for the parents of dependent children to apply for additional financial support under Schedule 1 of the Children Act 1989 but any such award is made for the benefit of the child only, and they are very narrow in terms of scope e.g. for school fees.
If you and your partner rented together and you are not named on the tenancy agreement, then you will have no right to remain in the house. The same applies if one of you owns any property, only the owner is legally entitled to stay there.
If you jointly owned a house with your partner, then you should speak to a family solicitor to understand the true value of your interest and the options for selling or buying out the other person.