lots of back lash on the carriage is normal for most lathes, can't say for sure about yours. It is just because the you are driving a rack and pinion rather than a lead screw. For really fine adjustment, you can either bump the hand wheel with your palm, bump the carriage with a mallet, or use the compound - use whatever method is best for the moment. You can get adjustment much finer than the lathe will actually cut by using the compound. Just set the compound to 60 degrees from the z-axis and you get 0.0005" advance on the z-axis for each 0.001" advance on the compound. Makes it easy to dial in that last 2 or 3 thou off a facing cut.
For a 3jaw, anything under 0.005" is fine. Most of use are happy if we get to 0.003", some of use live with something closer to 0.010". If you need better, use a 4jaw, spend huge money on a set true chuck, or loosen the chuck to adaptor screws and knock it around with a mallet. Best to just assume that anything in the 3jaw needs to start over size and will be off center. The same goes for a collet BTW on your machine, it will run better than a 3jaw, but not as well as a 4jaw.
The collet style you have pictured is far preferable to a draw bar style imo.
They are simple to make, buy a collet nut, and machine the rest from semi-hard steel. Unless you are planning to really use it hard, it can be made from semi-hard and not require hardening/grinding. It is an excellent project to practice turning to size, cutting threads, and setting up and turning tapers. Your shop made tool will actually be better than a commercial tool since it will fit the spindle to a higher tolerance and allow you to remove the work in the chuck for milling, and put it back on the lathe for finishing with better repeatability. All you need is a dial indicator and some marking fluid or transfer dye to test the fit.
Check out Myfordboy and Tomstechniques on YouTube, they both have excellent videos on setting up the compound for a taper. The $ is better spend on something else, put the cash in a jar for your new mill fund
-Josh