Very few mathematicions these days wish to base their mathematics on ZF without the axiom of choice, as your question seems to imply. Yes, there is an intuitionist school about which I don't know a whole lot, but they seem to be quite the minority. So let's consider ZFC. I think the main philosophical argument for it is that the axioms seem obviuosly true, if you are willing to believe that such things as infinite sets exist in some fashion, and that nobody has been able to come up with an equally obviously “true” statement about sets that is not a consequence of these axioms. The independence of the continuum hypothesis doesn't seem to bother people much, since to most of us it doesn't seem either obviously true or obviously false. Though some people are working on settling it by finding other axioms that will at least be considered likely true or at least useful. There was a recent article in the Notices about these efforts, in fact. But is there “hard evidence” that ZFC is the best we can come up with? Not by most mathematicians' standards I think, but there seems to be plenty of soft evidence.
I might add that many working mathematicians (I am talking here mostly about analysts, since they are the people I know best) don't care one whit about these questions, but happily go about their business using Zorn's lemma whenever it seems necessary and never let it bother them. And there are some who would rather base all mathematics on category theory rather than ZFC set theory, but that is not my cup of tea, so I will leave it unstirred.
(I think that the question of why planar graphs are exceptional is important. It can be asked not only in the context of graphs embeddable on other surfaces. Let me edit and elaborate, also borrowing from the remarks.)
Duality: Perhaps duality is the crucial property of planar graphs. There is a theorem asserting that the dual of a graphic matroid M is a graphic matroid if and only if M is the matroid of a planar graph. In this case, the dual of M is the matroid of the dual graph of G. (See this wikipedia article).
This means that the circuits of a planar graph are in one to one correspondence with cuts of the dual graph.
One important manifestation of the uniqueness of planar graphs (which I believe is related to duality) is Kasteleyn's formula for the number of perfect matchings and the connection with counting trees.
Robust geometric descriptions: Another conceptual difference is that (3-connected or maximal) planar graphs are graphs of convex 3-dimensional polytopes and thus have extra geometric properties that graphs on surfaces do not share.
The geometric definition of planar graphs (unlike various generalizations) is very robust. A graph is planar if it can be drawn in the plane such that the edges do not intersect in their interiors and are represented by Jordan curves; The class of planar graphs is also what we get if we replace "Jordan curves" by "line intervals," or if we replace "no intersection" by "even number of crossings". The Koebe-Andreev-Thurston theorem allows to represent every planar graph by the "touching graph" of nonoverlapping circles. Both (related) representations via convex polytopes and by circle packings, can respect the group of automorphisms of the graph and its dual.
Simple inductive constructions. Another exceptional property of the class of planar graphs is that planar graphs can be constructed by simple inductive constructions. (In this respect they are similar to the class of trees, although the inductive constructions are not so simple as for trees.) This fails for most generalizations of planar graphs.
A related important property of planar graphs, maps, and triangulations (with labeled vertices) is that they can be enumerated very nicely. This is Tutte theory. (It has deep extensions to surfaces.)
It is often the case that results about planar graphs extend to other classes. As I mentioned, Tutte theory extends to triangulations of other surfaces. Another example is the fundamental Lipton-Tarjan separator theorem, which extends to all graphs with a forbidden minor.
The study of planar graphs have led to important graph theoretic concepts Another reason (of a different nature) why planar graphs are exceptional is that several important graph-theoretic concepts were disvovered by looking at planar graphs (or special planar graphs.) The notion of vertex coloring of graphs came (to the best of my knowledge) from the four color conjecture about planar graphs. Similarly, Hamiltonian paths and cycles were first studied for planar graphs.
Graphs on surfaces and other notions generalizing planarity. Considering the class of all graphs that can be embedded in a given surface is a natural and important extension of planarity. But, in fact, for various questions, graphs embeddable on surfaces may not be the right generalization of planar graphs.
David Eppstein mentioned another generalization via the colin de Verdier invariant. This describes a hiearachy of graphs where the next class after planar graphs are "linklessly embeddable graphs". Those are graphs that can be embedded in space without having two disjoint cyles geometrically link. As it turned out this is also a very robust notion and it leads to a beautiful class of graphs. (They all have at most 4v-10 edges where v is the number of vertices; The known case of Hadwiger's conjecture for graphs not having K_6 minor implies that they are all 5 colorable.) Further classes in this hierarchy are still very mysterious. Other extensions of planarity are: 3) (not literally) Graphs not having K_r as a minor; 4;5) (Both very problematic) As Joe mentioned, graphs of d-polytopes, and also graphs obtained from sphere packings in higher dimensions; 6) (not graphs) r-dimensional simplicial complexes that cannot be embedded in twice the dimension, 7) [A notion that I promoted over the years:] graphs (and skeleta) of d-polytope with vanishing second (toric) g-number, and many more.
Forbidden minors and coloring. As for the second and third items in the question. About coloring I am not sure if we should consider 4-coloring planar graphs and coloring graphs on other surfaces as very related phenomena. Regarding forbidden minors. Kuratowski's theorem on surfaces is a special case (and also an important step of the proof) of a much more general result (Wagner's conjecture proved by Robertson and Seymour) about any minor-closed class of graphs. This result can be regarded as extending Kuratowski theorem and also (and perhaps more importantly) extending Kruskal and Nash-Williams theorem on trees. Indeed Kuratoski's theorem is related nicely to the more general picture of topological obstruction to embeddibility. If you would like to propose a different (perhaps topological) understanding of the extension of Kuratowski's theorem for surfaces, then maybe you should start by the well-quasi ordering theorem for trees.
Best Answer
A few reasons for the importance of planarity having little to do with the need for maps:
A matroid is both graphic and co-graphic if and only if it is the graphic matroid of a planar graph
Planar graphs are the graphs with Colin de Verdiere invariant ≤ 3. As such they form a sequence with the trees, outerplanar graphs, planar graphs, and linklessly embeddable graphs.
The graphs of three-dimensional convex polyhedra are exactly the 3-connected planar graphs (Steinitz's theorem).
A minor-closed graph family has bounded treewidth if and only if it does not include all the planar graphs.