because Henry say he could
Taxation of the church.
Henry 2 fall out with Becket because instead of helping Henry Becket Began to stand up to him and defend the right of the church.
I don't think there is any real evidence indicating that Becket hated Henry II. On the contrary, it looks much more like Becket liked Henry. The problem was not hatred, but the fact that Becket had been given a job with importance to the church, and part of that job included protecting the church and the interests of the church. When Henry sought to acquire those interests for himself, Becket was duty bound not to let him. Becket was stubborn, but he seems to have been acting quite honestly and dispassionately.
Becket stood up against Henry II for the rights of the church against the demands of the state, one of many classes between the church and the crown. It was a precursor of Henry VIII's break with the church.
Becket was ordained and consecrated as Archbishop of Canterbury (and therefore head of the Church in England) on 3 June 1162. The Pope was overall head of the Church in western Europe, including England.
Becket ran away to France because he excommunicated every one in the church who was loyal to Henry II.
Because he wanted all the church to his side
By excommunicating with the church
To control the church courts
Catholic and then the Church of England when Henry left the church. Everyone was Catholic in his time.
Because he supported Reginald de Broc and the king in the quarrel with Becket. The feud was not Henry II vs the Church, it was Henry II and various members of the church, including the Archbishop of York and the Bishops of London and Salisbury vs Becket.
Thomas Becket refused to sign the Constitutions of Clarendon which restricted ecclesiastical privileges and curbed the power of the Church courts and the extent of Papal authority in England.