Personally, I wouldn't recommend lace yarn for socks. I think it would wear out very quickly and come out of shape also,...
As to the thickness, it depends. Lace weight refers to a wide variety of thicknesses in yarn. You can have "thicker" or "thinner" laceweight. It's a broad term.
As to the needles, you use much larger needles than you would need for the thickness of the yarn to get a lacier, airier finished garment. I have never tried knitting lace on small diameter needles, and probably never will, but you would need very thin needles.
The ways of Yarn are inscrutable.
I'm NOT a "miss", I'm a lady!