OK, I will also give it a shot.
First of all, I don't like to sell Geometrization because it helps with the homeomorphism problem. The Geometrization Theorem is an object of stunning beauty ("most 3-manifolds are hyperbolic" should be an exciting statement for anybody in an elevator who has seen the art of M.C.Escher), and beauty in mathematics is usually a sign that we are on the right track. And indeed, the beauty of Geometrization begets all kinds of results.
Regarding the new results of Agol, Wise et al. one should perhaps not jump right to "virtual Haken" or "virtually fibered" but one should look at the "real theorem",
the Virtually Compact Special Theorem which goes as follows:
If $N$ is a finite volume hyperbolic 3-manifold, then $\pi_1(N)$ is virtually compact special, i.e. $\pi_1(N)$ is virtually a quasi-convex subgroup of a Right Angled Artin Group (RAAG).
One can explain a RAAG to anybody who has seen group theory in an elevator between about 3 floors. The fact that "simple" objects like RAAGs contain all hyperbolic 3-manifold groups (up to going to a finite index subgroup) is stunning and beautiful. All the goodies, e.g. largeness, linear over $\mathbb{Z}$, virtual fibering, LERF, virtually biorderable etc come from that statement (well, together with Agol's fibering theorem, tameness etc.).
This can be seen clearly by looking at Diagram 4 in a recent survey paper on 3-manifold groups by authors whose names escape me at the moment. It is really stunning how the Virtually Compact Special Theorem answers all open questions at once.
It is one of the great achievements of Dani Wise to have found the "right statement".
(Note that largeness, linear over $\mathbb{Z}$, biorderable do NOT follow from virtual fibering or virtual Haken alone.)
Back to the elevator:
The results make me think that hyperbolic 3-manifolds are like Jack in the Box.
If you take a hyperbolic integral homology sphere you look at a tiny manifold, but when you press a button (i.e. go to an appropriate finite cover), the 3-manifold suddenly becomes a grand object of beauty (e.g. has as many fibered faces in the Thurston norm ball as you could wish).
(This analogy also works with tiny seed, a bit of water, blooming flower etc. for the botanically minded elevator companion)
So to conclude, I think the Geometrization Theorem and the Virtually Compact Special Theorem of Agol-Wise are stunningly beautiful results. The fact that the statements are so beautiful made it highly plausible that they were right, even before they were proved (I can't imagine that any serious person doubted the Poincare conjecture after Thurston stated the Geometrization conjecture). And ideally it's this beauty which I would like to communicate.
I think the answer to the first question is yes and the answer to the second one is no:
Yes, the quotient is an orbifold. The action of the finite group $G_x$ in a neighbourhood of $x$ can be linearized (at least if the action is by diffeomorphisms, I don't know about $C^0$ regularity), and the quotient $M/G$ is locally modelled on $G_x \backslash T_xM / T_x (G\cdot x)$.
No this orbifold is not good in general. For instance, you can glue a solid torus with a trivial circle fibration to a solid torus with a Seifert fibration with one singular fiber in the center and get a closed Seifert 3-manifold with one singular fiber. The fibration is given by the orbits of an action of S^1 and the quotient orbifold is a sphere with a single orbifold point, the simplest example of an orbifold not covered by a manifold.
More generally, I think every orbifold $M$ if dimension $n$ is the quotient of a manifold $P$ by an almost free action of the orthogonal group $O_n$ ($P$ is the principal $O_n$-bundle associated with the orbifold tangent bundle equipped with an orbifold Riemannian metric).
Best Answer
I believe this is worked out very nicely in "Geometrization of Three-Dimensional Orbifolds via Ricci Flow" by Bruce Kleiner, John Lott (http://arxiv.org/abs/1101.3733).
An atlas for an $n$-orbifold $\mathcal O$ consists of a Hausdorff paracompact topological space $|\mathcal O|$ together with an open covering $\{U_\alpha\}$, local models $\{(\hat U_\alpha,G_\alpha)\}$ ($U_\alpha$ connected open subset of $\mathbb R^n$) and homeomorphisms $\varphi_\alpha:U_\alpha\to \hat U_\alpha/G_\alpha$ satisfying a compatibility condition. An orbifold is then defined by an equivalence class of such atlas. (See page 6 of Kleiner-Lott.)
A smooth map $f:\mathcal O_1\to\mathcal O_2$ between orbifolds is given by a continuous map $|f|:|\mathcal O_1|\to |\mathcal O_2|$ with the property that for each $p\in |\mathcal O_1|$, there are local models $(\hat U_i,G_i)$ ($i=1$, $2$) and a smooth map $\hat f:\hat U_1\to \hat U_2$ equivariant with respect to a homomorphism $\rho:G_1\to G_2$ such that $\pi_2\circ \hat f = |f|\circ \pi_1$ where $\pi_i:\hat U_i \to U_i$ is the projection ($\rho$ is not required to be injective or surjective). (See page 7 of Kleiner-Lott.)
I think this satisfies your requirements and is in the spirit of Thurston.
Edit: Perhaps I should mention Remark 2.8 in the Kleiner-Lott paper (also in regard to other answers to this post), which recalls that an orbifold can also be seen as a smooth proper étale grupoid (and Morita-equivalent grupoids correspond to equivalent orbifolds). A grupoid morphism gives rise to an orbifold map, but these correspond to a stricter class of maps called good maps. The advantage of these maps is that one can pull back orbi-vector bundles.