Your "fiance's" children are his legal next of kin. If he dies intestate they will inherit his property under the state laws of intestacy. If he writes a will and leaves property to you, the will must meet any statutory requirements for disinheriting children. Otherwise the children may be able to make a claim. You have no other legal rights unless you are married.
what would be any legal reason preventing my fiance and I from living together
No. If the fiance is the natural father, then he does have rights to his son. Getting married doesn't give him legal rights to your child (assuming he is not the father). He must adopt in order to have legal rights.
No. Although next of kin is an inexact term depending on the context, a fiance is not next of kin. An unmarried adult's (who has minor children) children and parents would be considered next of kin. See related question link. A fiance would have no legal rights unless the decedent had executed a document granting legal authority.
I'm not a lawyer but I don't believe he would have any legal rights to it.
You need to ask clearer, your question makes no sense. What rights are we talking about?
You have no legal rights in your 'fiance's' property. If he dies then his property will go to his next-of-kin, his children, according to the laws of intestacy in his state. He could leave the property to you in his will, however, in some jurisdictions the adult children can claim a share. In most jurisdictions a person cannot disinherit minor children by will. You can check your state laws of intestacy at the related question link below.
see link below
No, not unless the fiance's parents consent and they would most likely have to give up rights to you and your father so they are no long liable for their child
Only if the non-custodial parent give up his parental rights.
yes 19 rights for children
No. A thirteen year old cannot execute a "legal" document.
Action on Rights for Children was created in 2001.