How and why did the Battle of Britain affect the US?
The US, among other countries not involved with the European war, were spectators to the Battle of Britain, which took place in the fall of 1940. The only special effect the battle had upon the US; was the fact that Britain and the US speak a common language, and the US had at one time been part of the British Empire (Revolutionary War 1776). In effect, it was like watching our relatives involved in a fight.
The Battle of New Orleans increased American self-confidence and increased the fame of Andrew Jackson, helping to lead to his election as President in 1828, but otherwise had no effect on the country. The peace treaty had already been signed when the Battle began. The outcome of the Battle had no impact on the terms of the treaty or the lands that the US held or that Britain lost control of.
What officially ended the American revolution and forces Britain to recognize the US as an independent country?
The battle of Yorktown was the last major battle, and that's what ended the war. Representatives from Britain and the States met in Paris and signed the Treaty of Paris, which said that Britain will acknowledge the US as an independent country and that they will remove all troops from US soil. It also set the boundaries for the US, among other things.