1. Yes, with single GPA registered in your favour, you can transfer her title of the property in your favour or in favour of anybody. It will be applicable on all concerned in connection with the said property.
2. Since the property is at Pune the POA and also the deed of conveyance shall have to be registered at Pune.
3. Yes, the said GPA will stand as a valid instrument for you for dealing with her share of the jointly held property including executing and registering a relinquishment deed in your favour. It does not matter if the said Deed in original is with the Bank.
4. You should get it done at the earliest since she might decide to withdraw her said GPA. Moreover, in case of her demise, the said GPA becomes invalid instantly.
5. Your are right. The both the GPA and the Relinquishment Deed shall have to be registered at Pune. The GPA is also required to be registered since as per the Act, all documents including GPA, in connection with transfer of title of properties shall have to be registred.