Change of sur name after marriage

Sir, May I request you to have 2-3 minutes time to read this and give me a reply so it help me in future. History : My Name is MP Renuka Kiran from birth certificate to all certificate and all other documents such as Adhar, PAN, Voter ID card and Bank account. My Husband’s name is S Madhu Babu. I married in 2009 as Hindu and give birth to a male child in 2010. Our marriage was held in village and not registered with any authority. At the time of applying for a birth certificate my husband had quoted my name as S Renuka Madhu babu instead of MP Renuka Kiran. While producing this birth certificate in school for his admission, they observed my name mentioned in birth certificate does not match with my ID proofs. While applied for change of name in birth certificate the GHMC authority asks me to get gazette notification. After seeing so many blogs and notes from you I would like to know te following:- 1. Is it mandatory to register our marriage as per Hindu marriage act ? 2. Is there anywhere mentioned in the act that I should change my surname after marriage. 3. Can I quote Bomby High court order dated sep 2011 regarding this to issue a certificate by GHMC? Note: From 2013 we are living separately and not likely to rejoin/divorce in near future. I Humbly request you to give a valid reply with the authority so that I can get my child admitted in a good school without changing all my ID proofs. Thanking you.