Ditto, not familair with a Lang lathe. Some pics would be nice, so we can see what we are talking about.
That said, yes .003" movement at the headstock spindle is at least .0025" too much. More like about .003" too much. There is absolutely no point at all in proceeding any further with alignment etc until you have fixed the headstock bearings. This amount of slack would be enough to create the taper on your test turning piece. So you may not have a bed wear or bed alignment problem at all, just a headstock bearing adjustment problem.
Depends on what the shims are like, but usually you can cut out new shims from shim stock and insert them in place of the old ones. How thick are we talking about? If they are big enough, you might be able to very carefully file them down, using a mike to make sure they come down parallel. Or you may be able to rub them on emery cloth on a sheet of glass to knock a few thou off. It does not take much to remove the .003" you seem to require.
EDIT: PS just had a squiz at Lang Junior on the lathes.co.uk site. It looks like if you have the roller bearings, there would be a large circular shim that goes under the collar that provides bearing pre-load? You should be able to cut a new one out of shim stock that is a bit thinner than the original and try it out?
The website also indicates you have a very good quality machine there "better than Dean Smith and Grace and many German machines" is pretty high praise. So well worth persevering with, cautiously so as to not overdo things.
I'm not a big fan of spending money on ground test bars. You can turn one up yourself between centres if you sort out that headstock bearings and then the tailstock alignment. But even then, final adjustment is best done by turnin a 6" long piece of bar as you have done. This gives you the "real world" reading of how the lathe is cutting, which takes into account wear in the carriage and cross slide etc and the forces on them under cutting load. Static alignment is one thing, working alignment another.
It seems unlikely that a carriage crashing into the headstock would budge it on a solid machine like this. I woiuld not worry about that possibility too much.
You need to measure the length and the diameter at each end of the headstock spindle taper. It may not be Morse, but some odball like Brown and Sharpe or Jarno or a couple of others that used to get used. Or it could be a Morse 5-1/2 which is a MT5 but only half the usual length.
Edited By Hopper on 24/04/2017 09:23:06