Typically the taper on a barrel starts after the thread tenon, which is cylindrical. Most gun barrels are made from 4140 CM, or 416R SS. I use a 6 jaw adjust-tru chuck, and either a spider or a reverse center on the outboard end.
After I spent many years single pointing threads, I purchased a geometric die head to do the barrels that were in high demand. With a geometric die head it only takes 2 passes for a finished thread, at 30 seconds a pass. It would normally take me 12-15 min to single point one.
Class 3 is a production standard, but some like them a little tighter depending on if it's a switch barrel or not.
To do this many barrels, I have 3 lathes set up for various operations. I use SB 13" for turning and threading, a Jet 1340 for chambering, and a SB heavy 10" set up with a belt sander for polishing.