You can find a variety of opinions about the Battle of New Orleans, but I think it goes as OTL provided the American commander is sufficiently aggressive. The British were not equipped or prepared for a real fight, never mind a siege; they expected a walkover, and didn't bring enough ammo. If the Americans make a sortie on Dec 23 as per OTL, the British get slow and cautious, and by the time they're ready to attack the line, said line has been fortified beyond the British ability to take. I don't think it takes a military genius to realize that early sortie is a good idea.
Without Jackson, no one exceeds orders in 1819, so Florida remains Spanish.
1824, let's assume Calhoun stays in the race rather than take the safe spot of VP. He and Crawford still split the South and JQA wins in the House. But Calhoun will not be VP nor in the Cabinet - can he take the Presidency in 1828 from outside?
The British had no intention whatsoever of taking Louisiana for themselves. They intended to give it back to Spain. The Americans will be taking it from Spain by force even if the British win; a loss would discredit the Madison administration pretty badly, but I'm not sure a Federalist resurgence is really possible at this point - they're pretty moribund.