Approximation of Wasserstein distance between $p_\theta$ and $p_{\theta + d\theta}$ Given a parametric family of distributions $\{p_\theta\mid\theta \in \Theta\}$, one can show that under some regularity conditions, the following approximation is valid
$$\operatorname{KL}(p_\theta\parallel p_{\theta + d \theta}) = d \theta^TF(\theta) \, d\theta + \mathcal O(\|d\theta\|^3),
$$
where
$$F(\theta)_{ij} := \mathbb E_{x \sim p_\theta}\left[\frac{\partial^2}{\partial \theta_i \, \partial \theta_j} \log(p_\theta(x))\right]
$$
is the Fisher information matrix of $p_\theta$. A very rough sketch of the proof can be found on wikipedia.
Question 1
Is there such an approximation formula for the Wassertein distance or other measures of discrepancy between probability distributions ?
Question 2
Same question, specialized to $f$-divergences (of which KL is a particular case).
 A: With regard to your first question, one such result shows that under some conditions on $q$, the following inequality holds for all p:
$$W_2(p,q) \leq \sqrt{\frac{ KL(p\parallel q)}\rho}$$
This is a so called Talagrand$(\rho)$ inequality, which holds whenever a suitable log-Sobolev inequality holds for $q$. For more information, the paper of Otto and Villani [1] is a good reference. This is all contained in Villani's book on optimal transport as well. This also contains many other inequalities between the various distances and divergences, so it is a good reference. However, due to the work of Otto, the Wasserstein 2 metric induces a formal Riemannian metric on the space of densities, and its geometry can be quite different from that of the Fisher-Rao metric.
As to your second question, the Fisher metric is unique in that it is the quadratic term of the Taylor series for any $f$-divergence. For more information on this phenomena, see [2]. 
[1] Otto, F., & Villani, C. (2000). Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. Journal of Functional Analysis, 173(2), 361-400.
[2] The Fisher Metric Will Not Be Deformed https://golem.ph.utexas.edu/category/2018/05/the_fisher_metric_will_not_be.html
A: A natural field here is Wasserstein information geometry. 
See Wuchen Li, Guido Montufar: Natural gradient via optimal transport
https://arxiv.org/abs/1803.07033
For related applications see my talk
https://speakerdeck.com/lwc2017/learning-via-wasserstein-information-geometry
