Did Jackson support states' rights?
Although the quick answer may appear to be "no," Jackson in fact had a strong states' rights record: his handling of the Georgia-Cherokee problem is one example. What he opposed was states extending themselves into federal business in a way that could threaten the Union, as was the case with the Nullification Crisis.
Although Jackson won the popular vote in the 1824 Presidential race, he did not win the Presidency. What prevented him and why?
Jackson did not win more than half of the electoral vote, which threw the race into the U.S. House of Representatives. There, each state–large and small–has only one vote. New York tipped the race in favor of John Quincy Adams.
What part of his own experiences helped motivate Jackson in the Battle of New Orleans?
The American Revolution against Britain had left Jackson an orphan and claimed both of his brothers as well. For the rest of his life he harbored a severe dislike of the British–especially when they were on American soil.