  For a planar self-affine set satisfying the strong separation condition, it
has been recently proved that under mild assumptions the Hausdorff dimension
equals the affinity dimension. In this article, we continue this line of
research and our objective is to acquire more refined geometric information. In
a large class of non-carpet planar self-affine sets, we characterize Ahlfors
regularity, determine the Assouad dimension of the set and its projections, and
estimate the Hausdorff dimension of slices. We also demonstrate that the
Assouad dimension is not necessarily bounded above by the affinity dimension.
