Q1) how did british conquer india and establish their rule?
The British had come to India as traders. However, their ambition kept growing and so started the fortification of the port cities. After establishing its monopoly in Indian trade after removing all competing European traders, the English aimed for imperialism. The gradual interference in Indian politics helped the British to win the Diwani rights of Bengal. Later following several instruments such as subsidiary alliance and imposing cultural superiority of British culture over the orient East, they reduced India to a weak but resourceful colony.
The colonization of India by Britain started when the East India Company was formed by some British traders, who saw an opportunity in India and took it. To store the goods they had they built many warehouses, which gave them an excuse to build forts and build up armies to "protect" them. Britain saw how well India was in the trading business, and wanted to take control of it to have all the trading posts for themselves.