Probating the will and mutation of property

My father died last month leaving behind a three-story building and some bank balance. In his registered will, of which I am the Executor being the eldest son, he has devised one floor to each of us three (two brothers and one sister). On the floor meant for my younger brother, who is now no more and died even when the father was alive, his wife lives with his family. Likewise, my sister also occupies one floor and lives with her family. The floor meant for me has been let out for the time being since I live elsewhere. My questions: i) How do I go about getting the property vis-a-vis individual floors mutated? 2) Is it necessary and advisable for me to get the will probated? 3) Some banks have been asking me to bring accession certificate from the court to get some un-nominated money left behind. The money duly nominated has been released to me which I have since distributed among the beneficiaries. How do I get the said accession certificate without getting the will probated? Is it necessary or advisable for me to get the will probated? 4) I understand getting the will is not mandatory in Delhi, to get the properties mutated and moreover, it is time-consuming and expensive to get it done. 5) What is the benefit for getting the will probated and how much would it cost? Are we also required to get individual floors registered in our individual names after the will is probated? Please advise. Rajendra Prasad