We compute the Hausdorff dimension of any random statistically self-affine Sierpinski sponge $K\subset \R^k$ ($k\ge 2$) obtained by using some percolation process in $[0,1]^k$. To do so, we first exhibit a Ledrappier-Young type formula for the Hausdorff dimensions of statistically self-affine measures supported on $K$. This formula presents a new feature compared to its deterministic or random dynamical version. Then, we establish a variational principle expressing $\dim_H K$ as the supremum of the Hausdorff dimensions of statistically self-affine measures supported on $K$, and show that the supremum is uniquely attained. The value of $\dim_H K$ is also expressed in terms of the weighted pressure function of some deterministic potential. As a by-product, when $k=2$, we give an alternative approach to the Hausdorff dimension of $K$, which was first obtained by Gatzouras and Lalley \cite{GL94}. The value of the box counting dimension of $K$ and its equality with $\dim_H K$ are also studied. We also obtain a variational formula for the Hausdorff dimensions of some orthogonal projections of $K$, and for statistically self-affine measures supported on~$K$, we establish a dimension conservation property through these projections.