First, is there a certain reason why Napoleon was exiled to Elba? Why Elba?
Second, if it didn't have to be Elba, AND if the Louisiana Purchase never occurred, could it have been plausible that Europe would have exiled Napoleon to Louisiana?
I didn't really think Napoleon had a say in where he was exiled. The only reason why I thought of Louisiana is because the War of 1812 had just ended and maybe the British would have sent Napoleon West to make him America's problem (not that he would be able to do that much with just Louisiana).
First, is there a certain reason why Napoleon was exiled to Elba? Why Elba?
Second, if it didn't have to be Elba, AND if the Louisiana Purchase never occurred, could it have been plausible that Europe would have exiled Napoleon to Louisiana?
If the sale of Louisiana never occurred, then it would still be a French possession in 1814. I can't imagine Louis XVIII wanting Napoleon to live in his territory.
I think it's more plausible to have him take exile there if it is a U.S. possession.
Actually Elba was chosen by Alexander I, without any consultation with the other allied powers. It really angered both Franc and the allies that he was sent into exile so close to France and Italy. Talleyrand wanted St. Helena, which I guess he eventually got. The Azores and Madeira were also mentioned. Also, Marie Louise was the one made ruler of Parma, not Empress Josephine.
You're right about Marie Louise, sorry I got his first and second wives switched. However, where are you getting the Alexander I part from? As you can see above, I put word for word the treaty article, and it does seem to have, at least for the purposes of the treaty, made to look like Napoleon was fine with it. And really, the only thing I can find is that Britain was upset, nothing about other nations being upset about it so close to France and Italy.
As I read the wording of the treaty, it does seem to have been of Napoleon's choosing, though I don't know the backstory if he suggested that specific island, was given a list of different places, or it was a compromise. Exact wording- "The island of Elba, adopted by his Majesty the Emperor Napoleon, for the place of his residence, shall form, during his life, a separate principality, which shall be possessed by him in all sovereignty and property."
Even if you consider that the tonnage New Orleans port would handle would be less in ATL where it was not bought by the US, it would still certainly be second only to NY in outgoing tonnage (in OTL New Orleans surpassed NY in outgoing tonnage until 1825 and the Erie Canal opened). With vast reaches and more Americans (like Daniel Boone and family) coming to Louisiana to settle there's quite a bit of potential to that territory.
The British were opposed the treaty (and the representative there refused to put his signature on it) because they believed Elba gave Napoleon too much power and potential. That's why the British agreed for their warships to constantly patrol around the island. Blockading Lousiana halfway around the world is going to be more of a resource drain. Not to mention a threat to Canada! The British were so Napoleon paranoid that they'd be stationing a huge standing army in Toronto!