Firstly, sir, it is with regard to a each and every specific state in terms of their own rules only in regard to the OBC where the state plus central act keeps in mind for determining the eligibility criteria.
Secondly, though as per the law with respect to issuance of SC and ST certificates, it is only father on whose basis the child may get the benefit.
Thirdly, and yes if the father is having the certificate in X state then same would be given to the child but not in other state.
Fourthly, by making an application you can get that mistake corrected in the SDMs office.
Fifthly, for getting admission in schools and colleges there is no stringent law which says that you can’t get admission in the state for where your certificate doesn’t hold value.
Sixthly, it is only made once in a lifetime but you can take copy of the same if lost it.
Rest you can consult me through Kanoon for detail discussion over phone, so that I may also be able to help you with the making of the application also.