$\mathcal{M}et(M)$ 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)

In particular, for the Sobolev order $\ge 2+\frac{\dim(M)}2$ 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.

More details:

In the $C^\infty$-topology, where $\mathcal{M}et(M)$ is an open subset of a Frechet space, Q2 is has always a positive answer. See here for a detailed description of this topology. Namely, the mapping $g\mapsto R^g$ which maps a metric $g$ to its curvature, is smooth $C^\infty$, and thus continuous since all is Frechet. Then, choose a finite open atlas $U_\alpha$ for $M$ a compact $K_\alpha \subset U_\alpha$ such that $\bigcup_\alpha K_\alpha = M$, and a frame $(s^i_\alpha)$ on each chart. Then, if for all $i < j$ sectional curvature $$ k(\text{span}(s^i_\alpha,s^j_\alpha)) = -\frac{g_0(R^{g_0}(s^i_\alpha,s^j_\alpha)s^i_\alpha,s^j_\alpha)}{g_0(s^i_\alpha,s^i_\alpha) g_0(s^j_\alpha,s^j_\alpha) - g_0(s^i_\alpha,s^j_\alpha)^2} \ge \epsilon_\alpha $$ for each $\alpha$, then we have $>\epsilon_\alpha/2$ for each $g$ near $g_0$. This also holds for the $C^2$-topology, or for the Sobolev metric of the order given above, involving the Sobolev lemma.