For your first question, Sachs proved that every linklessly embeddable graph on $n$ vertices has at most $4n-10$ edges (this is not true, see comments).
The answer to your second question is no. This follows because Apex graphs are linklessly embeddable, and one easily checks that they do not have bounded genus.
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.
