Power chuck?
Is it possible the noise is introduced from the power chuck system, which you no doubt have ?
To check this out reduce the chucking pressure, as well as change from OD chucking to ID chucking and see if this effects the noise. It may be a part of the axial system for actuating the check, and this stuff is not really hard to service once the case is off the headstock area.
As for the taper, I just compensate in the program by adding turning a small taper on purpose that just cancels any error. I have been too lazy to adjust the tailstock !
I can't measure any taper once compensated for by careful adjustment of finish cut.
Super X3. 3600rpm. Sheridan 6"x24" Lathe + more. Three ways to fix things: The right way, the other way, and maybe your way, which is possibly a faster wrong way.