Your question (1) may not be true as stated. One approach that I like is to think of the product of two $G$-CW complexes first as a $(G\times G)$-CW complex, which works because $G/H\times G/K \approx (G\times G)/(H\times K)$. Then you can consider it as a $G$-space by restricting along the diagonal $G\to G\times G$. However, in "Restricting the transformation group in equivariant CW complexes," Sören Illman showed that the restriction of a $G$-CW complex to an $H$-space does not generally give an $H$-CW complex. He then gives a construction of an $H$-homotopy equivalent $H$-CW complex with nice properties.

Related, and related to your question (2), is Illman's paper "Existence and uniqueness of equivariant triangulations of smooth proper $G$-manifolds with some applications to equivariant Whitehead torsion." One consequence of this paper is that $G/H\times G/K$ has a $G$-triangulation, so a $G$-CW structure. The problem that arises when you look at a product of two $G$-CW complexes is that the triangulations of the products of orbits don't have to interact well with the original CW structures.