Yes, there is my question. And whether a complete non-compact Alexandroff spaces (with n-dimensional Hausdorff dimension or n-dimensional topological dimension) with curvature bounded below by 0 and Euclidean volume growth is homeomorphic to $\mathbb{R} ^n$?

The question askes for the statement in the last part of your answer.

Futhermore, does it imply that the metric is isometric to the Euclidean metric up to scaling? Whether it also holds for complete non-compact Alexandroff spaces (with n-dimensional Hausdorff dimension or n-dimensional topological dimension) with curvature bounded below by 0?

Yes. Can it be helpful if we add the condition: $|v|_g\geq |v|_{g_{st}}$ for any $v\in TS^n$?

Thanks. Do you have higher-dimensional examples? Since I need the existence to attack the problem of scalar curvature. In particular, whether the map $Id: (S^n, g)\to (S^n, g_{st})$ is a harmonic map? Here $Id$ is the identity map, $n\geq 5$ and the scalar curvature of $g$ is positive.