  We show convergence of the gradients of the Schr\"odinger potentials to the
Brenier map in the small-time limit under general assumptions on the marginals,
which allow for unbounded densities and supports. Furthermore, we provide novel
quantitative stability estimates for the optimal values and optimal couplings
for the Schr\"odinger problem (SP), that we express in terms of a negative
order weighted homogeneous Sobolev norm. The latter encodes the linearized
behavior of the 2-Wasserstein distance between the marginals. The proofs of
both results highlight for the first time the relevance of gradient bounds for
Schr\"odinger potentials, that we establish here in full generality, in the
analysis of the short-time behavior of Schr\"odinger bridges. Finally, we
discuss how our results translate into the framework of quadratic Entropic
Optimal Transport, that is a version of SP more suitable for applications in
machine learning and data science.
