Since the property of being linklessly embeddable is a minor-closed property, one can apply the graph minors theory of Robertson and Seymour.  From the [Graph Minors Structure Theorem](http://en.wikipedia.org/wiki/Graph_structure_theorem), graphs in a fixed minor-closed family have a linear number of edges.  That is, there is an absolute constant $C$, such that every linklessly embeddable graph on $n$ vertices has at most $Cn$ edges.  This is a vast generalization of the fact that planar graphs on $n$ vertices have at most $3n-6$ edges. 

**Update.** In this paper of [Sachs](http://link.springer.com/chapter/10.1007/BFb0071633), it is proved that every linklessly embeddable graph on $n$ vertices has at most $4n-10$ edges.  

The answer to your second question is (most likely) no for 2-manifolds.  This essentially follows from the necessity of apex vertices and vortices in the Graph Minors Structure theorem.  That is, for a general graph $H$, the [genus](http://mathworld.wolfram.com/GraphGenus.html) of graphs without an $H$-minor is not bounded.