wedlock

Dear scholars, As-Salamu `alaykum. In the Name of Allah I bear witness that none is worthy of worship other than Allah and that Muhammad is His servant and final Messenger. To proceed: I have two children from before I accepted Islam and they were both born out of wedlock in an unlawful relationship before I met my wife. Since accepting Islam I gained custodial rights over them but not custody. They live with me and my wife who also have two children and of course I care for and maintain them. I call them son (6) and daughter (5) and they call me father. My wife calls them step-son and daughter and they call her step-mother. I have learned that children of zina are not ascribed to the father and that there is no parent/child rights between them. They are considered orphans and there is no term as step-parent. Having said all of this back ground info., my question is what are the relations between us according to Shari`ah? Should I allow them to know they are orphans? If their birth mother wants them back though she refuses acceptance of Islam should I allow them to leave my home to live with her? If I am not considered their father than what is our duty to them and them to us? Jazakum Allah khayran.