Is every metric uniformly close to a metric with negative scalar curvature?
Jochen Lohkamp answers your first question in "Curvature h-principles", Ann. of Math, vol 142, p 457–498. If $\dim M\ge 3$, then the space of negative scalar curvature metrics is dense in the $C^0$ topology. More remarkably, he even shows this for negative Ricci curvature metrics.
About the $C^0$-closure of metrics with nonnegative scalar curvature, the results actually holds without any completeness assumptions and thus can be applied locally, so the fact that your manifold has boundary doesn't make any difference here. There are now two proofs available :
The theorem was first proved by Gromov in Dirac and Plateau Billiards in Domains with Corners, it proceeds by showing that non-negative scalar curvature is equivalent to the non-existence of cubical domains with :
- $(1)$ mean convex faces.
- $(2)$ angles between adjacent faces less than $\pi/2$.
This is accomplished by gluing several (regflected) copies of such a cube and smoothing the resulting metric to get a metric with positive scalar curvature on the $n$-torus, which do not exist thanks to previous work (Gromov-Lawson, or alternatively Schoen-Yau in dimension $n\leq 7$).
To prove $C^0$ stability from this, one consider a sequence $g_i\xrightarrow{C^0} g$, and assume that $g$ has negative scalar curvature at a point $x$. Around such an $x$ one can build a small cubical domain $C$ with the two properties above, the hard work is then to show that for $i$ big enough, one can slightly deform $C$ to get a cubical domain $C_i$ which has the same properties $(1)$ and $(2)$ with respect to $g_i$, which is hard because the positive mean curvature condition at first glance looks like a $C^2$ condition on the metric.
The second proof is (for me) much simpler and uses Ricci flow, it was given by Bamler in A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature.