I finally got the Casati all dialed in, strung up the derailleurs and had some trouble shifting into the lowest gears, but eventually got it. Now, originally I had problems with the chain skipping and heard that I should move the wheel back a little in the dropouts, so I went ahead and pushed it all the way back. Kept skipping, so I replaced the freewheel with an almost NOS Regina. Since it was STILL skipping, I ordered new jockey wheels from Soma. Got it all re-done today, and guess what? My chain skips in the middle cogs on the freewheel.
My only current theory is that the chain is just the tiniest bit too narrow. It's an 8-7-6-speed Sram chain. Should I look into getting a(nother) new chain and seeing if that solves the problem? Any other ideas?