The product rule for a dot product, a cross product, and a vector dotted with itself all carry over from ordinary calculus in the expected way: d/dt(A·B) = A·(dB/dt) + (dA/dt)·B, d/dt(A×B) = A×(dB/dt) + (dA/dt)×B (keeping the order, since cross product isn't commutative), and A·A = |A|² by the definition of the dot product. All three hold.

By the defining property of the cross product, A = B×C is always perpendicular to both B and to C individually -- that part is guaranteed. But B and C themselves can be at any angle to each other (the cross product is defined regardless of the angle between them) -- there's no requirement that B and C be perpendicular to each other.