1. If my children have officially been adopted by my new husband, then will my children have rights on their deceased father property & their former grandfather ancestral & self acquired property?
No. They cannot have any rights in the properties of their biological father side. They can claim rights from their adoptive father only.
2. If my new husband does not adopt my children officially, in this case if their former grandparents file a petition/case in court to get custody of their grand children as their new father does not adopt officially and looking for their bright future, can court give rights custody of children to former grand parents?
They are your children and not your husband's.
Hence you have all the rights to reject their claim for custody of the children.
For any reason the court will not pss any order to hand over custody of children to their grandparents during your lifetime until and unless the children interest and welfare and their life is at at stake if they live with you.
3. Please advise my rights in following property:-
• Rights in deceased husband property
• Rights in ancestral property
• Rights in former father in law’s self acquired property.
Since you have remarried, you dont have any rights and NO is the answer to your above questions.