Registering a property in Mother's name

My father and mother were residing in a 500sq ft apartment in Mumbai which is part of a society registered in 1975. My father passed away in 2020 without a will. My father was the first owner and had shares alloted in his name. He had made my mother the nominee to the property. My mother continues to reside there and has now been made a member of the society through his nomination. We now wish to sell this property as my mother wants to relocate and she intends to buy a property at the new location. My brother (who lives abroad) & I have no issues with this and can sign an NoC or POA to effect the registration. Questions are - 1. Does this property have to be registered in my mother's name before selling it or can we do it at the time of sale directly to the buyer i.e can a nominee sell the property? 2. If we want to register the property in mom's name right now, are registration and stamp duty charges applicable, considering it is a property passed on from my father to my mother? I have read that there have been some changes to stamp duty charges for old properties and hence they are negligible. Also what would be the approximate registration charges? 3. Is there any need to register the property if we don't intend to sell it? TIA Priya