"Q1 Where should I start in ensuring that everything is actually running true?"
Easiest way to start off with is to chuck up a piece of aluminium the length you say you will be working with, no tailstock centre, and take a cut over the full length. Measure it and see how the lathe is cutting – parallel or tapered. If you get lucky and it's turning parallel, no need for further mucking about with headstock or bed.
To check the tailstock, mount the piece of aluminium between centres and take a fine cut over it. Measure the bar after cut and if its tapered, adjust the tailstock to suit until it turns parallel. Double check for vertical wear/misalignment by bringing the tailstock and headstock centres together and put a steel rule in between. It should stand up vertical if alignment is ok.
Then, at least, you have a starting point.
Before you do the above, the lathe is best bolted down to the bench. Preferably a steel bench, as solid as possible. When bolting the lathe down, check all four feet with feeler gauges. If you can slip a feeler gauge under a foot, slip a piece of shim of that thickness under the foot. This way, when you tighten the four mounting bolts down, they will not pull the bed out of shape. Again, this provides a starting point where you know where you are at. You can add more shims later IF it turns out the bed requires levelling.
Otherwise, it can be very easy to spend a lot of time chasing your tail aligning one part of the lathe then another, which throws out the first and so on.
Your .7mm runout on the chuck is way excessive by a factor of about 10 or so. You should look at what is going on there before doing any test turning. Check the runout on the actual spindle. Then on the chuck backing plate. Then on a known round object held in the chuck, such as a gudgeon pin or at least a good piece of round bright mild steel. That way you can determine if the problem lies in the spindle, the backing plate or the chuck jaws themselves. Runout of the actual body of the chuck does not necessarily tell you much.
Edited By Hopper on 07/09/2016 14:07:18
Edited By Hopper on 07/09/2016 14:15:54
Edited By Hopper on 07/09/2016 14:17:33
Edited By Hopper on 07/09/2016 14:18:15