In the Civil War era you had the following processes available:
investment casting (brass and steel)
sheet metal rolling
sheet metal stamping
That's more than enough to build an AK-47.
Ferrous metallurgy and heat treatment weren't an exact science back then; you picked the ore and the smelting process, selected a foundry to match, and worked out the heat treatment by guess and by gosh and experience. All lost arts now, but a production engineer of the 1850s wouldn't have needed long to sort things out.
Deep drawn brass cases were known at the time. The suggested .36 caliber would require less of a bottleneck and would probably work better with black powder.
As I mentioned earlier in the thread, analyzing and duplicating a nitrocellulose-based smokeless powder would have been beyond the chemistry of the 1860s. Nitroglycerine was known at the time, and had any rounds loaded with Cordite been available, it would have been simple enough to analyze and duplicate them. However, unless someone specially loaded some x39 with Cordite and gave it to them as a sample, they'd use black powder.
The mercuric primers of the time would work. Analysis and manufacture of chlorate primers would have been within their capability.
Having read "The Guns of the South" (which wasn't really one of Turtledove's better efforts) I think the part about the Confederacy developing their own AK was just some throwaway padding in a sluggish story that was already repetitious and slow. However, whether by diligent research or by accident, he was right - it was within the capability of the Confederacy to do it.
The South Africans had brought in AKs and ammunition by the containerload. Davis and Lee began to wonder about their ally's motives, and whether the supply of arms might be cut off, which is what led to the "Confederate-made AK" scene. But Lee had plenty of AKs by then; all he really needed to worry about was a continuing supply of ammunition, so making their own AKs made no sense given the scenario. On the other hand, why should the Confederate government have had any more sense than the Federals...