These graphs are the incidence graphs of cubic graphs, a.k.a. 2-stretches of 3-regular graphs.
I'll write $I(G)$ for the incidence graph of $G$.
Given a graph $G$ and an integer $k$, it's NP-complete to determine if $G$'s crossing number is at most $k$ (i.e., whether $G$ can be drawn in the plane with at most $k$ edges crossing each other), even if $G$ is restricted to be cubic. Clearly, the crossing number is not affected by adding an extra vertex in the middle of each edge. (Source: Hlineny, "Crossing number is hard for cubic graphs", J. Combin. Theor. B 96(4):455–471; DOI.)
It's possible that the bandwidth problem for these graphs is NP-complete, since it is NP-complete for trees where every vertex has degree at most three. (Source: problem GT40 in Garey and Johnson for general graphs; for low-degree trees, Garey, Graham, Johnson and Knuth, "Complexity results for bandwidth minimization", SIAM J. Appl. Math. 34:477-495; Citeseer.)
Various NP-complete graph problems remain so on cubic graphs and these lead to NP-complete problems on the corresponding incidence graphs that are reasonably natural. For example, asking if a cubic graph $G$ has a dominating set of size at most $k$ is equivalent to asking if $I(G)$ is a union of at most $k$ copies of $I(K_{1,3})$. Likewise, an independent set in the cubic graph corresponds to a set of disjoint copies of $I(K_{1,3})$ in $I(G)$.