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 by C. Villani (2003).
   For instance see Open Problem 7.20.

2. A geometric study of Wasserstein spaces: Euclidean spaces 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 by L. Ambrosio and N. Gigli (2012).
   For instance see Open Problem 5.7.

4. { Euclidean, Metric, and Wasserstein } Gradient Flows: an overview by F. Santambrogio (2016).

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 by C. Villani (2003).
   For instance see Open Problem 7.20.

2. A geometric study of Wasserstein spaces: Euclidean spaces 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 by L. Ambrosio and N. Gigli (2012).
   For instance see Open Problem 5.7.

4. { Euclidean, Metric, and Wasserstein } Gradient Flows: an overview 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.