For turning wooden balls, get a bit of steel tube and make sure the cutting end is square and sharpen it like a leather punch with the taper on the outside, use this tool after you have roughed out the ball using the whole front edge to cut. To get a number of uniform sized balls, just make then the size of the bore of the tube, when the ball is to size it just pops into the tube.
Turning wood in the metal lathe with metal turning tools, hard, fine grain wood is good(mature Rimu in NZ), just don't use soft new Radiata Pine, you'll have a real hard job even on a wood lathe with the sharpest tools.
Ian S C