Just stop your feeding for a second to break the chip. Those rat's nests can be dangerous!
Turning between centers is a better way of dong test bar cuts. A chuck may produce some forces that distort the test part. Put a dead center in the head stock. You can leave the chuck on , just back the jaws away from the work. The chuck jaws can be used to drive the lathe dog. It takes less material length if you use a face plate to drive the dog instead of the chuck.
Ideally you shout turn the taper on the dead center while it is mounted in the spindle. That ensures that it is running concentric. I still don't under stand shimming the tailstock?? A few .001" up down has very little affect on the diameter turned. Not measurable with the means you likely have.
If you are feeling the need to check everything, make sure the head stock is aligned with the travel of the carriage. You should find the recommended method in the paperwork that came with the lathe. A headstock misalignment will cause a taper to be turned on a bar just held by the chuck. It will also affect the test cuts when the work is held by the chuck and the tailstock. That's why you should do test cuts for tailstock alignment between centers. You will need a bar about 2 1/2" in diameter x 12" long, sticking out 10". Lock the compound and cross-slide to make the cut. If both ends of the cut don't measure the same your headstock needs aligning.