$\mathcal{M}et(M)$ carries many natural (= invariant under the action of the group of diffeomorphisms of $M$) Riemannian metrics. See the following papers (and references therein):

- Martin Bauer, Philipp Harms, Peter W. Michor: Sobolev metrics on the manifold of all Riemannian metrics. Journal of Differential Geometry 94, 2 (2013), 187-208. [(pdf)][1]

In particular, for the Sobolev order 2 metric the curvature is continuous so Q2 has a positive answer. 

A more quite recent paper concentrating on the well-posedness of the geodesic equations for Sobolev metrics is [this one][2].


  [1]: https://www.mat.univie.ac.at/~michor/rie-met2.pdf
  [2]: https://www.mat.univie.ac.at/~michor/rie-met3.pdf