I don't see why the CSA wouldn't survive.
It had more resources than most nations, a cadre of well educated leaders, and good ports. I think Lee would almost certainly become the second President and would do well depending on his health. Like the US, the next 30 years of presidents would be soldiers so perhaps a President Johnston, Breckinridge, Gordon, Early, Rodes, as the grays get grayer.
Its largest city, New Orleans was aimed to the south and was a great trade city. I could see it becoming the trade hub of the Caribbean/ Gulf of Mexico and could see the CSA being the dominant state of what is now Latin America either through influence or even territorial expansion.
The CSA and USA would share most issues of foreign affairs, mainly keeping Europeans out of the New World. I could see friendly relations much like Canada and the US.
Slavery could end around when Brazil ended theirs, probably a gradual ending as of anyone born after a particular date is no longer a slave.