A) The common name for the British Indian Empire was the "British Raj''
Derived from local languages, Raja meant King and Raj was meant to denote a Kingdom. After the British East India company gave the control of the country over the to British Throne, the area was then known as British Raj, or the Dominion of the British.
The British were the first power in history to consolidate power in as as far flung places as Aden in Yemen to Ragoon in Burma and develop a single government system around it calling it the British Raj.
The First capital was in Calcutta but it was later moved to Delhi in the West.
With regards to British influences in India, A) the common name for the British Indian Empire was the British Raj. The British East India Company ruled India until the rebellion in 1867 which then brought direct rule from the British Government.