At what conditions a compact metric space of covering dimension $n$ (on $\mathbb R^n$) is an n-manifold?

Question

In the discussion to the MSE post an answer of @MatthewPancia with correction in a comment of @JasonDeVito would state:

Every compact metric space of covering dimension $n$ can be embedded isometrically into $\mathbb R^N$, with $N$ sufficiently large ($N$ grows roughly as an order of $n^2$)

1. It sounds like basically any such metric space is actually a manifold. In a sense that it can locally be approximated by a vector space, which approximation would provide it with a natural manifold structure.
   Is it? May be not for the whole space, but for its sub-spaces of covering dimension $n$? May be not for any such space, but at some conditions on metrics?

2. Let now narrow it down to Riemannian metric spaces on $\mathbb R^n$. What are conditions for them to be manifolds, smooth manifolds, smooth manifolds everywhere but countable set of sub-spaces of zero measure?

3. How about Finsler metric spaces on $\mathbb R^n$ to be manifolds in this sense?