1. Only legal remedy is via legal adoption and that also before child attains ‘age’ of h/er majority as per prevailing law. It is simple procedure in District Court of child's ordinary residence and for the same hire any Junior level Advocate found via reference and provide to him legal marriage certificates of the lady with you and for rest the hired Advocate will assist.
2. The new couple (biological mother and foster father) has to seek consent of the biological father. if the biological father is providing alimony (child support, he may have a say. On moral grounds, he cannot object to the change is also one view.
3. However, once the mother has custody, the kids' last name is her decision. The law is not very strict about it.
There is no codified law, irrespective of the status of the marriage the children take on the father's second name. The biological father has the right to obtain a stay order if their second name is being changed.
On your 1 – Birth certificate mentioned biological father’s name cannot be changed in reference to query.
On your 2 – The advice which you already sought and placed here for verification is based on short sight and may lead to other Public Document issues if suppose child continuously not studying in future till Xth. class in same school where such advices made documents submitted hence previous Para 1 opinion is long term advice for all Public Documents purposes in future till child attains age of majority as per Law.