Thought I would drop back in. If the lathe is level, put a dead centre in the tail stock, then see if one can be put in the head stock spindle with out removing the chuck. Remove chuck if need be, gently run tail stock up towards head stock, when tips of centres are close to touching, lock down tail stock. Carefully advance ram in tail stock to almost touch tips of centres together, this is where we see what shimming tail stock feet did!! Tips should line up from every angle viewed at 90 degrees to the bed of lathe. If tips don't line up, maybe remove all shims and see what happens. This gives a base line to work from. It does not indicate if bores of head stock and tail shock are parallel to each other or to the lathe bed. There is usually some way to move tail stock left and right. This adjustment is used to line up tail stock or to misaligne for cutting tapers.( longer then compound will do).
This still does not show you where the tail stock centre will be in relation to to head stock centre at the far end of lathe bed. Time for a test bar. Test bar length will depend on what is at hand, it has to able to support it own weight over the length you want to check. A 1/2in. or 12 mm. bar over a 6ft/2 m. length will not work well. And you should be able to easily handle it. It looks like this lathe is 36in. between centres. I would start with a piece about like you had, looks like 1in., but longer, maybe 18 in.
This is one way to line things up or see what you have. Put test bar in chuck, a 3 jaw will do for this, have enough material out of the jaws to turn end square and centre drill , 3/16 centre drill or so. Also machine O.D. of bar about 1/2 to 1in. This dia. is fairly important because you want the same dia. on the other end of the bar also. Turn bar end for end, machine and centre drill again. Do centre drilling and O.D. maching on end without moving bar, before turning for other end.
Put centres back in, support bar between centres. Do not put a lot of pressure on tail stock ram to support bar. Just a nice gently firm turn of the hand wheel, you want to be able to turn test bar with fingers. A tiny drop of oil wil help on each end. Don't use a live centre.( a variable). Now I hope you have a dial gauge. A sharp pointer would work, a sharp lathe bit. Put dial gage on/in tool holder, put probe end on machined O.D of bar, zero, gently turn bar, should stay at zero for full rotation. Now for the big one, pull probe back away from bar, Do not move cross slide or compound!! Move carriage to tail stock end of bar, put probe on this machined O.D. Do we still read zero, turn the bar gently, should read zero. Horrah, great, dam I like this lathe.
When we don't read zero, can we live with it, is it a tenth or 2, or?... Does the reading change when turning the bar? Not centre drilled on centre or burr, chip in taper ofcenter drilling. Ram abit loose, centre moving around in centre drilling. If the reading does not change when turning bar but is out anyway, tail stock may need to be moved sideways, Or,Or, tailstock could be low or high.
This does not show whether bore of head stock or tail stock are out of line with lathe bed.
For a giggle or 2, put dial gauge at centre of test bar, zero dial, unlock ram of tail stock, gently turn hand wheel tighter, did the needle move? depends on dia. of bar. Don't get the cheater pipe on the hand wheel. Put dial gauge probe on body of tail stock by the ram if possible. Put a bit of pressure on with hand wheel again, tail stock moveing a bit? While dial gauge is at centre of test bar, gently push bar back and forth by hand, any different readings? If you put the gauge against head stock body put some pressure on, if this moves life may be about to get interesting.
Another long story, and this is only the very basics.