All properties follow directly from the definition equation 2, and the definition of dot and vector products. By the way, if the vectors A, B, C are constant the same rules apply as for ordinary vectors.
The best way to deal with such quantities is to drop the vector and vector product notation and work with the 3D fully antisymmetric Levi-Civita tensor $\epsilon_{ijk}$, which is 1 if ijk is an even permutation of 123, -1 if it is an odd permutation and otherwise 0. With this $$\nabla \cdot \left( \mathbf A \times \mathbf B \right) = \nabla_i \epsilon_{ijk} A_j B_k \,.$$ Summation over i,j,k is understood. A useful relation is $$\epsilon_{ijk} \epsilon_{ilm} = \delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl}$$.