Russia was one the strongest supporters of the Union during the war as it saw it as a counterweight to GB . The Crimean War was only about 4 years earlier than the ACW and the US was a historical enemy of GB. The US was clearly an up and coming power by that time being the 2nd most industrialized country per-capita and the 3rd(barely behind France) over all in the world. Russia had a huge population and army. Each could see each other as a counterweight to GB/France. Russia could use help in industrialization and the US could use some of Russia's military might.
Neither party has any viable way of projecting their political influence, much less perceived power in support of the other in the face of each others most likely perceived enemies....
At best they offer each other kind words and moral support but that's about it....There is nothing they cAN REALLY DO FOR EACH OTHER.
for the US they have a keener interests in keeping the cordial relationship that is now starting to bud between itself and the UK for economic reasons. They need British capital....
Also. while the "peculiar institution" that pervades the southern states persists...any thought of alliance in any form is anathema to British politicians, unless the North does something that appears overtly threatening. and deliberate on the part of US government policy. politically and economically the north is more aligned with Britain even if a budding junior competitor, but that's still far in the future. If anything this is likely to further the British hold on northern industries as Br. capital finds it more politically palatable to invest there than in the south. That does not mean there won't be investment in the south obviously where the economic dividend is apparent Br/French and all other European investment will follow in a private way...
Only the French would consider the possibility, but only out of necessity if they were to commit to supporting the Mexican venture. Which of necessity means no Franco-Prussian war as the French interest is diverted elsewhere
a kind of proxy war could end up being fought in Mexico if Nappy III is sufficiently committed to it, but I'm not certain that he would be....It was a huge financial drain for no real benefit. but French investment in Confederate reconstruction could be encouraged as a prid pro quo for Confederate support in propping up Maximilian. the whole venture might still fail of course as the US would then more actively aid the Juaristas..
Its difficult though to see either US successor state being interested in continuing their war by proxy after exhausting themselves on their own fields of battle
Spain at this time hasn't abolished slavery as yet....so while the relationship is not likely to be warm and rosy, it should be somewhat civil at least, unless the confederates keep pushing the Cuba button... because on their own it will never happen for them. Either by purchase or by invasion. okay never might be a bit strong...but well the probability is very low. unless Spain somehow gets bound up in the European alliance system.
In the immediate term I would not see either state seeking an immediate alliance partner but rather continuing the previous Unified American policy of isolationism where the Europeans are concerned. Private interests though will almost certainly establish themselves.