What is the maximum number of edges of an $n$-vertex linklessly embeddable graph?

A more general question is the following. What is the maximum number of edges of an $n$-vertex graph with Colin de Verdière number $\mu$?

A related question would be the following. Is there a fixed space (say a 2-manifold) such that graphs which embed linklessly into $\mathbb{R}^3$ are characterized by embedding into that space? This is purely out of curiosity.

Thanks.