The author continues with his study of multivariate smooth general singular integral operators over \(\mathbb R^N\), \(N\geq 1\), regarding their simultaneous global smoothness preservation property with respect to the \(L_p\) norm, \(1\leq p\leq \infty \), by involving multivariate higher order moduli of smoothness. Also, he studies their multivariate simultaneous approximation to the unit operator with rates. The multivariate Jackson type inequalities obtained are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function. In the uniform case of global smoothness, the author proves optimality. At the end, he lists the multivariate Picard, Gauss-Weierstrass, Poisson-Cauchy and trigonometric singular integral operators as instances of the general theory.