For your first question, [Mader](http://link.springer.com/article/10.1007%2FBF01350657) proved that all $K_6$-minor-free graphs (which includes all linklessly embeddable graphs) on $n$ vertices have at most $4n-10$ edges (thanks to David Eppstein for the reference).   

The answer to your second question is **no**.  This follows because [Apex graphs](http://en.wikipedia.org/wiki/Apex_graph) are linklessly embeddable, and one easily checks that they do not have bounded [genus](http://mathworld.wolfram.com/GraphGenus.html).

For your more general question, you can apply the graph minors theory of Robertson and Seymour.  That is, for *every* proper minor-closed class of graphs $\mathcal{M}$, there is a constant $C$, such that every graph in $\mathcal{M}$ with  $n$ 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. Most minor-closed classes will not have bounded genus.  This essentially follows from the necessity of apex vertices and vortices in the [Graph Minors Structure theorem](http://en.wikipedia.org/wiki/Graph_structure_theorem).