From what you have said, you have machined a piece of metal, and found it to be tapered.
Climbing aboard one of my hobby horses, this is possible, not because of any fault with the machine, but because of the way that it is mounted. Despite your sturdy bench, it is possible that having the machine bolted firmly to it, this is, in fact, causing one of your problems.
A lathe with a bed that is twisted, although perfect in all other respects, WILL turn a taper.
The way to check / overcome this is either to use a sensitive level; or to use the method advised, by Myford, and repeated in "The Amateur's Workshop" by Ian Bradley.
But solve one problem at a time. Sort out the spindle run out (presumably on the register where the chuck is to locate), and then find and correct any other problems, as they arise.
No three jaw chuck will hold work without a run out. A good example will give 0.005" or less. Bad ones worse, much worse in some cases.
The only way to hold work truly concentric, is to clock it in a four jaw independant chuck. This will give a run out depending upon your patience. (Another skill to master!)
I would not be absolutely confident, even, of a collet chuck, although work machined in a chuck, located in the Morse Taper of the Mandrel should remain concentric, as long as the workpiece is not disturbed in the chuck.
You will plenty of help and advice on how to solve a problem, on this site, (even if sometimes, views are divergent!)
Howard
Edited By Howard Lewis on 29/08/2017 16:44:13