That's a nice bench. I don't know what your plans are for the top, but it will need to be pretty stout to fully support your lathe without drooping. That model lathe is like 500lbs (IIRC). Criss-crossing the supports on both the top and the lower box mount would be wise.
It is important that the lathe sit level on a solid surface. Usually the lathe manufacture will give guidelines as to how they want you to mount it, and what the minimum thickness of material needed is (you probably already knew that, just pointing out the obvious to be...well, Cpt. Obvious).

Last thing you want is the ways drooping.
As for the box below, I'd mount it on a full length shelf from the bottom, with the same tubing you're already using (or stouter). I'd tie the legs and your support shelf all together too. For starters, the center of your tool box will be fully supported vs unsupported (that
may make opening the drawers difficult when everything is fully loaded down, but maybe not?).
Secondly, all four legs being tied together would help keep the top rigid and flat with the lathe mounted to it. You could even add gussets to the corners for additional support.
Just my 2 cents. Cool project!
Andy