For your first question, Mader 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 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.