There's no reason to think the South would collapse. It would eventually be dirt poor, but I think Southern nationalism from the victory would prevent it from ever being reabsorbed into the Union. After a while the people of the US would adopt a "good riddance" attitude, especially as the South's reputation goes down and down in Europe due to the continuance of slavery.
It takes way, way more than just "Southern nationalism" to make a viable nation-state. I should start by pointing out that without the large-scale migration of blacks northwards that occurred IOTL but wouldn't be allowed by the CSA, black population growth would outpace that of white southerners. They were already a third of the population of the South at the time of secession, and would have grown larger, which would inevitably lead to more and more violent repression to keep them all in chains. Think Sparta and the helots, for a sense of the long-term state of things. Also, there was significant Unionist support among white populations, especially in areas like Appalachia and Texas that could lead to (continued) counter-revolutionary guerrilla warfare should the CSA succeed. All of this in some pretty rough terrain, mind you.
Add to all this the fact that the Confederacy as designed was a statist oligarchy which forced non-slaveholders into the military while denying them a voice in government, along with the already poor economy that's poised to become completely unsalvageable in a few decades when the Boll Weevil strikes, and it really isn't a pretty picture. As for the Union letting them go: why should they? They control the Mississippi delta, a serious economic inconvenience, and are probably seen as the result of foreign meddling (the only realistic way for them to win independence in the first place). Put those together, and they're a threat to the American heartland, not to mention their proximity to the capital. No, I really don't see the US shrugging their shoulders, moving the government back to Philadelphia, and writing off the whole Mississippi south of Cairo, Illinois.
I suggest generally a stronger France at the expense of Germany, but not much stronger
Why? How would they benefit from this, one way or another?
1. Some European powers chose to back it, and as a result, the USA allies with other European powers. Result: The USA is MUCH more militarized in this timeline, and involved in European affairs. Britain has a BIG problem if they're seen as a friend of the CSA, as Canada is essentially ripe for the picking if a militarized USA decides to take it. It would take a lot of work to render Canada tough enough to deter the USA. A USA deeply involved in European affairs produces MAJOR butterflies.
Turtledove popularized this idea, but I don't see why it would ever happen. It just doesn't deliver any benefit to the European powers, since all the Confederacy has to offer them is a tool against the United States, but the US is only an enemy to someone in Europe if the latter chooses to make themselves America's enemy. It's a Catch-22. Funnily enough, though, if this had all happened in the 1890's, I think Kaiser Wilhelm would have been much more receptive, given his inexplicable desire to take on the US in the Caribbean for reasons that bewilder me...
2. Europe decides not to get involved in the USA/CSA feud--perhaps realizing that the CSA is doomed if the US decides to get serious about removing it from the board. Even in that situation, Europe needs to pay more attention to the USA than in OTL, as it has more military power. Europe may well, in that case, proceed much like in OTL for some time. Both sides in the European Alliance System will need to court the USA, or at least placate it...
I'm still in the camp that thinks a successful independent CSA requires British intervention, so that colors my assumptions regarding who the US would approach when shopping for allies. Russia is the first logical choice, since they also have a bone to pick with the Tommies, and given their good relations with Germany at this time, the latter's also an option. If the UK and France saw their relations cool, then the French might also get approached, but only if they didn't intervene jointly in the Civil War, which they might not have done, given they were already up to their neck in problems in Mexico. If we get more adventurous, then a more successful China might also prove an asset, since they and the US would probably share a dislike of both Britain and Japan.