Rugang Ye wrote two useful papers on this part of Perelman's work, doing the work carefully and clearly: doi.org/10.1090/S0002-9947-07-04405-4, doi.org/10.1090/S0002-9947-07-04406-6

When you replace $K$ by $\lambda$ you get (up to normalization) the integral of $\omega^m$, which is (up to normalization) the volume. (I think this is sometimes called Wirtinger theorem.) The particular constants $r$ and $(m-1)!$ are just normalizations depending on some conventions of how the various quantities are defined.

What's the intended purpose of the history-overview tag if not for questions like this?

Consider the case that $E$ is the holomorphic tangent bundle and $h$ is a Kähler metric. Then the two notions coincide.

It isn't fully clear to me how Ricci flow and minimal surfaces fit into the categorical setup you suggested, but there are many quite similar examples from differential geometry and PDE, such as Cheeger-Colding theory, mean curvature flow with surgery, harmonic maps, singular Kahler-Einstein metrics and conic singularities, singular Yamabe metrics, etc. I'm not sure how to judge the difference between singular space and singular object (eg the canonical singular minimal surface, the Simons cone, could be viewed as a smooth object on a smooth space), so maybe these aren't what you have in mind.

It certainly isn't obvious if you haven't seen it! Depending on your personal taste you can make it a special case of Stokes theorem, Green's identity, divergence theorem, or integration by parts.

Carlo Beenakker is correct except that what he wrote is called musical isomorphism. The differential (i.e. exterior derivative) of a function $f$ is a 1-form $df$, and the musical isomorphism defined by the metric $g$ turns this into the vector field $\nabla f.$ You can read for instance in John Lee's "Introduction to Riemannian manifolds" or most other introductory books on Riemannian geometry

Anyway, to answer the original question: as Bahri says in a footnote, Kuwert's thesis resolves the graphical issue with Schoen and Schoen-Yau's proof of the positive mass theorem, and the three-dimensional case is in Colding-Minicozzi's book. Moreover (not mentioned by Bahri), a device of Lohkamp's reduces the positive mass theorem to the study of scalar curvature on connected sums with the torus, and I believe that both Lohkamp and Schoen-Yau's claimed proofs of the higher-dimensional positive mass theorem deal with this context, so that the graphical issue does not even come up.

@MoisheKohan I don't think that's a fair read, since he refers positively to Colding-Minicozzi's alternative argument for the same point