Is the result of u × u = 0 perpendicular to both operands?

Yes, because 0 · u = 0, which means that they are perpendicular.

What about u × 0 ? The magnitude |u||0| sin θ is zero, and the result is a vector, so:

u × 0 = 0 × u = 0

Now, consider (ku) × v. Plug into the formula for the magnitude, |u||v| sin θ, and for the direction (perpendicular to both operands).

What is (ku) × v ?