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, 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.
The answer to your second question is 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 of graphs without an $H$-minor is not bounded.