Short answers, pulled from the story.
Nawab Diwan Murshid Quli Khan, also known as Kartalab Khan, began construction of the mosque between 1700 and 1704 CE. The Nawab intended for this building to serve as a place of worship while securing his own legacy.
This structure stands today in the Old City of Dhaka, Bangladesh. A high platform called a tahkhana supports the entire structure with rectangular rooms underneath that shopkeepers rent to generate income.
To the east of the platform stands a Baoli or stepped well which remains the only known example of its kind in Bengal. Archaeologists consider this feature to be of North Indian or Deccan origin since the builder had been in the Deccan before coming to Dhaka.
In 1777, control of the market passed to the daughter of Sharfaraz Khan who held the title Naib-e-Nazim at that time. Her name gave rise to the locality known as Begumbazar and some sources suggest the mosque itself may have taken its name from her rather than the original builder.
Mirza Golam Pir undertook extensive restoration efforts during the nineteenth century to preserve the structural integrity of the building for future generations. These repairs ensured the survival of the domes and the platform system without which the aging structure might have faced severe deterioration.