Is reemarriage with 10 year elder woman who has a 10 year old boylegal

Sir , I m 24 remarriage with 10 year older divorcee woman of 10 year old boy by Hindu marriage act . The name of Ex husband of my wife is not written in any of the documents of child and wife like Aadhar , driving licence and school certificate except birth certificate of child . Before first marriage girl was obc and that divorcee man was sc . . I only want to give my name to that boy in school without adoption​ legally I wanna know 1)when he will grow up with my name( without adoption ) can he claim my property legelly on behalf of my name in school ? . 2) His biological father( now divorced) was sc . and his first biological father Name is also written on birth certificate of child .can child claim the sc quota ? 3) Now there is no name of biological father in any documents of child only mother name is written . Can child claim mother quota ? ( Mother is obc )