The indestructibility and infinite length have, to the best of my recollection, never been tested by anyone other than Diana since Byrne changed the lasso (unless you count Donna running out of rope in the Heinboot, but, y'know... Heinboot). My default assumption has always been that without Diana, it can't be cut or broken, but that's it; the infinite length property would require semi-conscious control to be of any use, and be intimately related to her ability to twist the thing into weird, physics-defying flight paths during combat, which both speak to it being a Diana-specific ability.