This talk concerns sharp interpolation inequalities between weighted trace Hardy and Hardy inequalities. We improve these inequalities by adding Hardy-Sobolev type correction terms, involving superquadratic exponents of $u$ with Hardy type potentials of optimal singularity, covering the critical Sobolev exponent as well. A uniform consideration is presented yielding, as special limiting instances, optimal improvements of weighted trace Hardy and Hardy inequalities, the later extending previous results which correspond to the nonweighted case. It turns out that the trace Hardy and Hardy weighted inequalities share the same optimal Hardy-Sobolev type improvements. For the limiting trace Hardy case, we will also consider trace remainder terms. Such estimates have a special significance, as they can be translated into refined versions of fractional Hardy inequalities, via the characterization of the fractional Laplacian as a so called Dirichlet to Neumann map.