Partition of India is very bloody, so is Pakistan's suppression of East Pakistan. Here comes the questions:
Please note: for the purpose of this answer EP = East Pakistan WP = West Pakistan Pakistan = whole of Pakistan pre 1972
EP (during British India) stood at the frontline for the creation of Pakistan. Hussein Suhawardy was one the prominent leaders of the Muslim league and attached himself to Jinnah in the letters demand for a Muslim state. It wasn’t obvious that Pakistan would be unable to rule EP from the very beginning, or at least no more obvious than whether India or Pakistan would be able to rule themselves at all in the first place. The differences between EP and WP arose much later although Jinnah had sowed the seed for such differences as soon as the partition occurred by declaring Urdu as the national language completely ignoring the Bengalis of EP who were 55% of the population and had Bengali as their mother tongue. Despite having majority of the population EP was extremely poor and WP did little to help it, instead diverting their resources for their gain. Eventually, more than two decades of military rule after partition, when the very first elections were announced, they were won by Sheikh Mujib, a Bengali from EP. Pakistan establishment rejected the elections, arrested Mujib and this sparked a revolution in EP. WP unleashed a genocide in EP to crush the rebellion. They fought back, India intervened and supported the cause and finally Bangladesh was born.
So the British always wanted a united India because they wanted India to be a part of the commonwealth so the Indian army that they had raised could continue to fight their wars in the future. The British (and the rest of the West) was suspicious of Soviet expansion and felt that’s why they believed that a united India would be better equipped to fight this expansion. However seeing Nehru (freedom fighter and first Pm of India) and his party predominantly being a of the socialist and anti imperialist ideology, they realised that this would not be possible so they decided to shield India by an Islamic arch to prevent communism from infiltrating which they believed was likely considering the inequalities in India and the ideological leanings of its leaders.
Sources
The Great Game - Peter Hopkirk
Pakistan a Garrison State - Prof Ishtiaq Ahmed
The Blood Telegram - Gary J Bass