Properties of toroidal graph
Regarding (3), yes, 
if a graph $M$ has two vertex disjoint non-planar induced subgraphs $G$ and $H$, then $G\cup H$ (and hence $M$) is not toroidal.

I don't know a reference but here's a proof sketch.

Thinking of the torus as $T=S^1\times S^1$, if a non-planar $G$ is embedded without crossing on $T$ then its edges must meet every $S^1\times\{a\}$ and $\{a\}\times S^1$, or else we cut such a circle out and realize $G$ as planar. By tracing where they so meet we obtain a subgraph of $G$ realized as homotopic to some
$$\left(\{a\}\times S^1\right)\cup \left(S^1\times\{b\}\right)$$
(so two circles meet in $(a,b)$). Similarly for the second graph $H$ and say 
$$\left(\{c\}\times S^1\right)\cup \left(S^1\times\{d\}\right).$$
But then $G$ and $H$ are not disjoint (they share a vertex or have crossing edges) as the embedded graphs meet in $(a,d)$ and $(c,b)$.