The United States and Russia enjoyed a "curious entente" beginning in the 1840s. Their similar (and non-conflicting) expansionist courses, the rise of abolitionism, and dual antagonism with Britain were the major reasons for this "entente". The U.S. was benevolently neutral in the Crimean War and actively supplied raw materials, war supplies and volunteers to Russia.
In return, Tsar Alexander II ensured that this benevolent neutrality was reciprocated during the American Civil War. The efforts of Lincoln's Minister to Russia, the eccentric Cassius Clay of Kentucky, were well received. When Napoleon III proposed tripartite mediation of the conflict by Britain, France and Russia in October 1862, Russia rejected the proposal.
The most notable impact of the positive U.S.-Russia relationship can be seen in the Polish Crisis of 1863-64. When Poland revolted against the rule of tsarist Russia, it diverted the attention of the major European powers away from the American Civil War. France, Britain and Austria demanded a settlement on the principles of self-determination. As they did not back this up with action against Russia, Russia crushed Poland's revolt. Lincoln and Seward, out of political expediency, remained silent on the Polish revolt. Charles Francis Adams, U.S. Minister to Britain, noted that the Polish crisis had "done something to take continental pressure off us".
In the midst of the Polish crisis, Russia dispatched a squadron of 6 warships to New York in 1863. Russia had multiple reasons to do so in addition to demonstrating its positive ties to the United States: it wanted to prevent its entire fleet from being bottled up in the Baltic if war broke out over Poland, and it wanted to gather information about ironclad warfare and purchase warships.
And what was the U.S. reaction to the Russian squadron in New York? "For two months, three thousand Russian visitors attended parades, balls, and dinners, while U.S. bands played "God Save the Tsar" and toastmasters hailed Lincoln and Alexander II. It aroused further European concern of an alliance, eliminating any possibility of intervention in North America."
Positive U.S.-Russia relations continued after the American Civil War, and they played a helpful role in facilitating the purchase of Alaska in 1867. This era came to a close when Russian persecution of its Jewish population began to affect its relations with the United States in the 1880s and 1890s.
My source is From Colony to Superpower: U.S. Foreign Relations since 1776 by George Herring. Most of Herring's information concerning U.S.-Russia relations appears to be taken from Distant Friends: the United States and Russia, 1763-1867 by Normal Saul
Essentially, the two most important foreign powers during the Civil War were England and France. The English were initially highly supportive of the North due to their fervent abolitionist and anti-slave trade predisposition. However, this support waned over time for a number of reasons—not only because of Lancashire's suffering textile industry (which actually weathered Lincoln's blockade of the South pretty well), but also because England and the North had a tough time intimating what they believed and what their intentions were.
In particular, the Washington government had decided that they could not frame the Civil War as a war against slavery due to negative public opinion. Moreover, the Richmond government successfully convinced many aristocrats and newspaper editors that the South was the last bastion of pre-industrial, English-style culture being assailed by the aggressive, imperial North. So there was a fair amount of support for them, even though this support was predicated on the lie that slavery would undoubtedly be abolished in the South after it won its independence. Nevertheless, the English public seemed firmly in favor of the North—Southern attempts to drum up support in hard-hit areas failed, whereas many people attended pro-abolition meetings.
What's equally interesting is Americans' attitudes on the English during the war. There seemed to be a motherly semblance of the future "special relationship", but many Northerners (Irish immigrants and American newspapers in particular) were very suspicious of British intentions. Consequently, both countries at one point or another seemed convinced that they would soon be at war during the Civil War. The English feared that Seward would want to invade Canada in order to consolidate the country's losses and distract people from the secession. During the Trent affair, when a US ship kidnapped two Confederate diplomats off of an English ship, the American public and politicians were certain that Britain would not retaliate. But Lord Russell, foreign secretary, was drawing up plans to go to war over the issue until Charles Francis Adams, head of the US legation in London, admitted to him that Seward hadn't provided him with an official line and that he didn't seem to desire war.
Misunderstood intentions played a big part in this. Palmerston's government never wanted to go to war, but they held a (more or less when it came to shipping) strict policy of neutrality, which meant that they did not recognize the South as an independent nation. But that also meant that Lord Russell couldn't overtly help the North beyond trying to keep the French neutral as well, since the French emperor and his ambassadors were inclined to recognize the Confederacy. There were other problems, though, such as the fact that British authorities did a poor job (sometimes deliberately) of taking action against Confederate ships being built in British ports.
It's also important to note that news could take a long time to cross the Atlantic. British reporters would file their reports, which took maybe two weeks to get printed back home, and then those papers would take a further two weeks to arrive in the States. So Lincoln and his cabinet would be about a month late knowing what the English newspapers were saying about the war. Motions being made in parliament about strengthening the Foreign Enlistment Act (many Brits in the South joined or were forced into service despite this law prohibiting them from foreign service) or for recognizing the South's independence depended on what victories or defeats had recently been announced in the papers. Moreover, many reporters in the field had strong anti-Union biases, which led them to misrepresent the outcome of battles and to ignore just how awful of a situation the average Southern soldier was in. The spirit of the South was certainly admired, but it's troubles were substantially underplayed.
This comes from Amanda Foreman's A World on Fire: Britain's Crucial Role in the American Civil War (2010). I would highly recommend it for historians and laymen alike. It gives a great overview of the war, US foreign policy, and British politics during this time. Foreman gives a very detailed account of the attitudes and opinions of different politicians, ambassadors, soldiers, etc., and she offers a clear but nuanced picture of English sentiments about the Civil War.
You may read from time to time that the Vatican (or the papacy, depending on the source) recognized the Confederacy. Pius IX did write a letter to Jefferson Davis. In that letter, he did address Davis as the “[I]llustrious and honorable President of the Confederate States of America.” However, the full reality is not nearly so dramatic. This is the only place online that I have found the letter from Pius IX to Davis. It is a PDF, and it also comes from a less-than-stellar source. Ignore the rest of the content, just scroll down to the text if you want to read the letter. In sum, the letter is the equivalent of a form letter. It acknowledges receipt of Davis’ earlier letter, says that it would be really nice if there was peace instead of war, asserts that the pope will pray for peace, and that he will also pray that Davis will be granted grace.
That’s it.
So, why the address as “President?” Basically, the pontiff filled in the blank by looking at the signature of the original letter from Davis. With Italian unification and being a self-proclaimed “prisoner of the Vatican,” Pius IX had much greater concerns much closer to home than the Confederacy. Also, the idea that the pope would somehow back the South in the US Civil War is far-fetched, given the less than ideal way Catholics were treated in that region. So, if you hear someone claim that the pope recognized the Confederacy, refer them to this post so they can learn that the truth was much more banal—the pope just replied to Davis’ letter with bland pleasantries.