There is a very simple way to test and adjust the alignment of the Headstock
and Tailstock of any lathe.  Put a center in the spindle and tailstock.  
Move the tailstock up to where the two centers can touch and lock the TS down.
Place a double edge razor blade between the two centers and use the TS ram to 
run them together.  If they are perfectly aligned, they will hold the blade. 
If, not the blade will deflect to one side or the other or up or down.  That
will indicate which way the adjustment must be made.

The same process can be used to adjust cutting tools to the center line of 
the work.  Just run the tool against a blade and the work.  If it is held 
straight then the tool is centered.  If the blade tilts down then the tool 
needs to be raised, or vice-versa. Use 3/4" or + stock to set cutter height 
as any change needed is easier to make.

If you don't have a double edge razor anymore (like me) or don't want to risk
your fingers, you can use any thin springy metal.  I used a .004 leaf from my
feeler gauge set. A straight piece of metal banding would probably work as 
well.  I have used a small machinists straight edge in a pinch, but it is not 
as sensitive.

Unfortunately, the leaf I used today was so shiny that it doesn't show up 
too well in the photos.  So look close.  Photos are on the photo page under
Simple A...folder.