Product rule for gradient

Yes, the product rule as you have written it applies to gradients. This is easy to see by evaluating $\nabla (fg)$ in a Cartesian system, where

$(\nabla f)_i = \dfrac{\partial f}{\partial x_i}; \tag 1$

then we have

$(\nabla (fg))_i = \dfrac{\partial (fg)}{\partial x_i} = \dfrac{\partial f}{\partial x_i}g + f\dfrac{\partial g}{\partial x_i} = g(\nabla f)_i + f(\nabla g)_i; \tag 2$

since (2) holds for each coordinate variable $x_i$, we have

$\nabla (fg) = g\nabla f + f \nabla g. \tag3$

Yes you can. Gradient is a vector of derivatives with respect to each component of vector x, and for each the product is simply differentiated as usual.