@huscarle
In point of fact we, the US, received aid from the French, specifically from De Grass and Rocheambeau, during the later part of the American Revolution. Many U.S. American’s forget that in no small part are we deeply indebted to the French (and also by treaty to Austria) for our independence. Their Navy, in particular, proved to check the Royal Navy’s complete domination of the U.S. coastal cities. Had the French not provided over 8,000 men in the Siege of Yorktown (matching the number of US troops), in the final and most significant battle, it is uncertain that the outcome would have been the same. Of course the great irony of the war is that France’s support for American independence, by Louis XVI, wrecked the French treasury and ultimately contributed greatly to his overthrow, which cleared the way for Napoleons truly astounding rise to power.

The war of 1812, in the US / Canada, was really a rather pathetic opportunistic land grab in which the War Hawks & Madison were convinced that Canada would simply embrace the US as liberators and join us without a shot fired. From what I have read, we did not engage in direct discussions with Napoleon or the French government, certainly not in terms of direct military aid, and as you point out much of the war in the US was waged after Napoleons disastrous invasion of Russia. After Napoleons defeat, and the subsequent collapse of his Empire, I strongly doubt that France was in any position to send the United States any aid. In point of fact, after the Russian defeat, the US actually lost several battles, and the British successfully burned Washington DC once they were able to focus their full attention on us. I am of the opinion that had Britain not been thoroughly exhausted by War at that time, the US could have suffered significantly, as she did not yet have a strong federal system of taxation, and therefore could not raise funds or build the arms necessary to fight off a fully mobilized British Empire.

As for Napoleon and the Louisianan Purchase, the Little Emperor had realized that France could not realistically counter British sea power, and he knew that there was no way, without being able to counter the British at sea, that he could ever hope to hold the land claims of the Louisiana territories. Given that he wanted to finance his great campaigns for France’s dominance of Europe, I am sure that was the motivation to sell. I believe the comment about Napoleon believing he could conquer the United States after dealing with Britain is pure speculations, but it is consistent with his ambition.

Of course, in the US Civil War, the CSA was desperate for recognition of sovereignty by any European power – be they Britain or France. But both Nations populace were still living in the crimson shadow cast over their continent from the previous years of perpetual strife, and when they witness the unbelievably high casualty rates that were the result of the Amercan Civil War battles, due in no small part to the advancements in rifling and rifled cannons and the failure of tactics to meet the new technology, they wisely stood on the side lines. As @emperopenguin points out, Napoleon III toyed with the idea of recognizing the CSAs sovereignty, but he could not do so without British recognition as well, and by 1863 the outcome was close to certain. It is interesting that in this instance, Britain’s refusal to recognize the CSA was a significant factor that led to the forging of the modern United States.

With that said, I am curious as to how the rule book deals with the awesome and terrible technical advancements of warfare that were introduced during the civil war – the extended range of the rifled musket, and the lethal penetration power of the Minie (mini) ball. To have the same rule set apply to 200 years of war seems to disregard the many advancements that occurred during the period.