I work in a bit different field too, but your question is quite interesting for me. So I have looked up in the literature just for curiosity. Since there are (surprisingly) no answers yet, let me share some references where open problems related to Wasserstein space are mentioned:

1. [*Topics in Optimal Transportation*](https://www.xarg.org/ref/a/082183312X/) by C. Villani (2003).  
For instance see Open Problem 7.20.

2. [*A geometric study of Wasserstein spaces: Euclidean spaces*](http://www.numdam.org/item/ASNSP_2010_5_9_2_297_0/) by B. Kloeckner, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze,  Série 5,  Volume 9 (2010) no. 2,  p. 297-323. 

3. [*A user’s guide to optimal transport*](https://link.springer.com/chapter/10.1007/978-3-642-32160-3_1) by L. Ambrosio and N. Gigli (2012).  
For instance see Open Problem 5.7.

4. [*{ Euclidean, Metric, and Wasserstein } Gradient Flows: an overview*](https://arxiv.org/abs/1609.03890v1) by F. Santambrogio (2016).

In addition, I am not aware if an explicit formula for $W_p$ distance between two Gaussian measures is known for $p\ne 2$, see e.g. [this question](https://mathoverflow.net/questions/158731/1-wasserstein-distance-between-two-multivariate-normal).