1) Is it necessary to get the mutation done in my and my mother's name. If we do not get the mutation done, what are the negative consequences ?
Mutation is nothing but transfer of revenue records from the previous owner to the present owner of the property. Since your father is no more but the property still remain in his name in the revenue records, it becomes essential to transfer the same to the current owner. If you do not do it now, it may create problem at the time of disposing the property to a third person.
2) If mutation is to be done, Is it to be done at Ghaziabad Nagar Nigam (municipality) or GDA (ghaziabad development authority) or at the Tehsil (where property registration is done) ?
Mutation is generally done by the Revenue department, so apply for it in the Tehsildar Office.
3) Can I get the mutation done by myself or do I need to hire a lawyer ?
Not necessary, if you can understand the procedure, you can go ahead yourself.
4) What documents are required to be submitted for mutation ? Is it necessary to obtain the NOC from my sister even if registered will of my father is there ?
The original title deed, death certificate of the previous owner, registered partition deed in applicants' name, address proof, legal heir ship certificate and other documents that will be called for by the authority as per the circumstances.