Suppose that we have two compact Riemannian manifolds $(M,g)$ and $(N,h)$. Define the Gromov-Hausdorff distance between them in your favorite way, I'll use the infimum of all $\epsilon$ such that there are $\epsilon$-Gromov-Hausdorff approximations between $(M,g)$ and $(N,h)$.

Then define $d(M,N)$ to be the infimum of the Gromov-Hausdorff distance between $(M,g)$ and $(N,h)$ taken over all Riemannian metrics $g$ and $h$ with sectional curvatures bounded in absolute value by 1.

If you know $d(M,N)$ for various choices of $N$, what can you conclude about $M$?

I know that Cheeger-Fukaya-Gromov theory on collapsed manifolds exactly covers the case when $d(M,N)=0$; for example $d(M,pt)=0$ if and only if $M$ is almost flat. I'm interested in situations where these numbers do not vanish. For example:

If $\Sigma$ is some compact orientable surface how should $d(\Sigma,pt)$ depend on the genus?

If $S^n$ is the standard sphere of dimension $n$, to what extent do the numbers $d(M,S^n)$ determine $M$? Are there manifolds $M$ and $N$ for which $d(M,N)\neq 0$ but $d(M,S^n)=d(N,S^n)$ for all $n$?

Is there an $\epsilon$ depending only on dimensions such that $d(M,N)<\epsilon$ implies $d(M,N)=0$?

If anyone can point me toward a reference/paper it would be very appreciated.