Yes. Wagner's Conjecture/Robertson and Seymour's Theorem says that any graph family closed under taking minors can be defined by specifying a finite list of forbidden minors. For any surface $S$, the graphs embeddable $S$ without crossing edges forms a family closed under taking minors.

I haven't looked carefully at it but Jim Belk's introduction to graph minor theory seems good. On the linked page he mentions the following facts: the projective plane has 35 forbidden minors, the number for the torus is in the hundreds (the precise number/collection is not known), and in general the number of forbidden minors grows exponentially with the genus.

Yes. Wagner's Conjecture/Robertson and Seymour's Theorem says that any graph family closed under taking minors can be defined by specifying a finite list of forbidden minors. For any surface $S$, the graphs embeddable $S$ without crossing edges forms a family closed under taking minors.

I haven't looked carefully at it but Jim Belk's introduction to graph minor theory seems good. On the linked page he mentions the following facts: the projective plane has 35 forbidden minors, the number for the torus is in the hundreds thousands (at least, the precise number/collection is not known), and in general the number of forbidden minors grows exponentially with the genus.

Yes. Wagner's Conjecture/Robertson and Seymour's Theorem says that any graph family closed under taking minors can be defined by specifying a finite list of forbidden minors. For any surface $S$, the graphs embeddable $S$ without crossing edges forms a family closed under taking minors.

Yes. Wagner's Conjecture/Robertson and Seymour's Theorem says that any graph family closed under taking minors can be defined by specifying a finite list of forbidden minors. For any surface $S$, the graphs embeddable $S$ without crossing edges forms a family closed under taking minors.

I haven't looked carefully at it but Jim Belk's introduction to graph minor theory seems good. On the linked page he mentions the following facts: the projective plane has 35 forbidden minors, the number for the torus is in the hundreds (the precise number/collection is not known), and in general the number of forbidden minors grows exponentially with the genus.