About generalized planar graphs and generalized outerplanar graphs
In the notation of Lee and Streinu (citation below) the second class you list are the (2,3)-sparse graphs. They give an algorithm to test whether a graph is (k,l)-sparse in polynomial time. However, the situation with planar graphs and $|E'|\le 3|V'|-6$ is a little more complicated, because that inequality is not true for all sets of vertices (if it were true, no two vertices could be connected by an edge, because $3\cdot 2-6=0$). So the class of (3,6)-sparse graphs (in their notation) consists only of empty graphs.  Probably their algorithms can be extended to graphs for which the inequality holds for all sets of more than two vertices.

Lee, Audrey; Streinu, Ileana (2008), "Pebble game algorithms and sparse graphs", Discrete Mathematics 308 (8): 1425–1437, doi:10.1016/j.disc.2007.07.104, arxiv:math/0702129.