NO, Canada is part of Commonwealth of Nations. Queen is still head of state.

No, nothing of the sort is true. The US intended to spread the revolution (much like the French did in Europe) to their cousins in the northern British American territories. Eventually this idea, and tensions with the British led to the War of 1812. But relations with the British and Canadians cooled after the war.

Canada became independent through mutual consent of the UK and the Canadian people. Canada achieved responsible self government in 1867, after about 100 years of continual British governance. This was the first step toward independence, they had authority over all domestic policy, and consulted with the British over matters of foreign policy (for example, the Canadians were not all that happy about the Boer War).

Canada became independent in 1931, because the UK government, and the Dominion governments, felt it wasn't their responsibility to govern in any respect for the Dominions (the Dominions being Australia, Canada, Ireland, New Zealand and South Africa). So the British, along with the Dominion governments made each of them fully independent, creating the Crowns of Australia, Canada, Ireland (since abolished), New Zealand, and South Africa (since abolished).

However, as a constitutional quirk, given the Canadian Constitution was written in 1867, and written to make a non-independent nation, it was codified that the British must approve Constitutional amendments. Obviously this only applied as a ceremonial matter, and was not amended to remove this clause in 1982.