First, it is straightforward to adapt Haslhofer's proof to higher dimensions, using the Sobolev inequality: For any compactly supported $u$ such that $\|\nabla u\|_2 < \infty$, $$ \|u\|_{\frac{2n}{n-2}} \le C\|\nabla u\|_2. $$ I encourage you to try.

There must be a standard reference for this, but it is proved in Appendix B of my paper:

Convergence of riemannian manifolds with integral bounds on curvature. II Annales scientifiques de l’É.N.S. 4e série, tome 25, no 2 (1992), p. 179-199

First, it is straightforward to adapt Haslhofer's proof to higher dimensions, using the Sobolev inequality: For any compactly supported $u$ such that $\|\nabla u\|_2 < \infty$, $$ \|u\|_{\frac{2n}{n-2}} \le C\|\nabla u\|_2. $$ I encourage you to try.

If you google epsilon regularity, you’ll find several explanations of it, including anther Math Overflow questionn.

The beauty of this simple and elegant proof is that it does not require coordinates and the dependence of the constant in the inequality on the geometry of the manifold is purely through the Sobolev constant.

There must be other good references for this, but it is proved in Appendix B of my paper:

Convergence of riemannian manifolds with integral bounds on curvature. II Annales scientifiques de l’É.N.S. 4e série, tome 25, no 2 (1992), p. 179-199