The birth certificate and school record mentioned the name of your biological father, so the name of step father cannot be replaced. As far changing the name or sur-name is concerned that you can always do without any reason or ground, and just for your sweet will. You do at least in 2 national/local Newspaper publication of change of name/surname on the basis of affidavit duly nortarised about change of name/surname and get the change of name/surname done in the Gazette of India, copy of the Gazzete you get and use it. The Passport, Driving Licence etc. can get the change of name latter on by using these documents.
As far registration of the marriage under the Hindu Marriage Act,1955 is concerned that can be done anytime during the lifetime of both parties to marriage, proof of marriage like photographs, certificate of temple/priest or witnesses to marriage will be reqired. The parties can even marry once again according to Civil Form of Marriage under the Special Marriage Act,1954 and get marriage registered and marriage certificate under the Act. Proof of death of previous husband, the death certificate will be required in this case if you mention the martial staus of the woman as widow, but if she is just stated as unmarried under the column of marital status, no death certificate will be required.