For the groups $\Gamma$ in Conway-Norton, there is always a moduli problem of $\Gamma$-structures, but since the groups always contain $\Gamma_0(N)$ for some $N$, you won't be able to construct a universal family (because there is a $-1$ automorphism in the way). However, you will sometimes get a ``relatively representable'' problem in the sense of Katz-Mazur.
The upper half-plane quotients will be coarse spaces parametrizing objects of the following general form: You have a diagram of elliptic curves, with some isogenies of specified degrees between them, together with some data that tell you how much symmetry in the diagram you should remember. Since all of the groups normalize $\Gamma_0(N)$ for some $N$, the diagrams will typically involve cyclic isogenies of degree $N$ in some way, and the symmetrization will involve a subgroup of the finite quotient $N_{SL_2(\mathbb{R})}(\Gamma_0(N))/\Gamma_0(N)$.
The standard example is $\Gamma_0(p)^+$ for a prime $p$, which is generated by $\Gamma_0(p)$ as an index two subgroup, together with the Fricke involution $\tau \mapsto \frac{-1}{p\tau}$. The $\Gamma_0(p)$ quotient parametrizes diagrams $E \to E'$ of elliptic curves equipped with a degree $p$ isogeny between them. Taking the quotient of the moduli problem by the Fricke involution amounts to symmetrizing the diagram, so the $\Gamma_0(p)^+$ quotient parametrizes tuples $( \{ E_1, E_2 \}, E_1 \leftrightarrows E_2)$ of unordered pairs of elliptic curves, with dual degree $p$ isogenies between them. Equivalently, you can ask for a set of diagrams $\{E_1 \to E_2, E_2 \to E_1 \}$ where the maps are dual isogenies.
A less well-known example is the 3C group, which is an index 3 subgroup of $\Gamma_0(3|3)$, with Hauptmodul $\sqrt[3]{j(3\tau)} = q^{-1} + 248q^2 + 4124q^5 + \dots$. This group is labeled $\Gamma_0(3|3)$ in the Conway-Norton paper, because $\Gamma_0(3|3)$ is the eigengroup, namely the group that takes the Hauptmodul to constant multiples of itself. The 3C group contains $\Gamma_0(9)$ as a normal subgroup, with quotient $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. You can view the upper half-plane quotient as a parameter space of quadruples of elliptic curves, with a rather complicated system of cyclic 9-isogenies and correspondences that get symmetrized (more on this in the last paragraph). A more succinct expression follows from using the matrix $\binom{30}{01}$ to conjugate $\Gamma_0(3|3)$ to $\Gamma(1)$ and $\Gamma_0(9)$ to $\pm \Gamma(3)$. Then you're basically looking at a moduli problem that parametrizes elliptic curves $E$ equipped with an unordered octuple of symplectic isomorphisms $E[3] \cong (\mathbb{Z}/3\mathbb{Z})^2$ that form a torsor under the characteristic 2-Sylow subgroup $Q_8 \subset Sp_2(\mathbb{F}_3) \cong SL_2(\mathbb{Z})/\Gamma(3)$.
In general, you can encode moduli problems attached to arithmetic groups using the fact that congruence groups like $\Gamma(N)$ and $\Gamma_0(N)$ stabilize distinguished finite subcomplexes of the product of all $p$-adic Bruhat-Tits trees. Conway gives a explanation (that doesn't use the word "moduli") with pictures in his paper Understanding groups like $\Gamma_0(N)$. For example, when $N$ is a product of $k$ distinct primes, $\Gamma_0(N)$ stabilizes a $k$-cube. Given a finite stable subcomplex, there is a standard way to make a moduli problem out of it by assigning elliptic curves to the vertices, isogenies to the edges, such that the induced transformations on the Tate module behave as you would expect from traversing the product of buildings. To symmetrize, just enumerate orbits of the transformations you want, and demand a torsor structure.
In the case of the 3C group in the above paragraph, $\Gamma_0(9)$ pointwise stabilizes a subgraph of the 3-adic tree that is an X-shaped configuration spanned by 5 vertices. The edges coming out of the central vertex are in noncanonical bijection with points in $\mathbb{P}^1(\mathbb{F}_3)$, and to symmetrize, you can make an unordered 4-tuple of diagrams of 5 elliptic curves, related by the action of the subgroup $V_4 \subset PSL_2(\mathbb{F}_3)$ that preserves the cross-ratio.
Your $F_N$ is the functor people would usually mean when they talk about the functor classifying elliptic curves with full level N structure (though it's a bit nicer if you replace $(Z/nZ )^2$ with $\mu_n \times Z/nZ$, so that the determinant takes values in $\mu_n$ on both sides.)
Wikipedia is not wrong. (Wikipedia is surprisingly seldom wrong!) Your F_2 is indeed not representable by a scheme, which is to say that the scheme known as Y(2) is not a fine moduli space. To say it was a fine moduli space would be precisely to say it represents the functor in question. That's why Y(2) doesn't have to have a universal family over it.
Yes -- the stack is the same thing as the functor; but we're keeping track of certain facts about that functor when we say it's a stack. For that matter, in the case where the functor is representable by a scheme, the scheme is also the same thing as a functor; but in that case we don't think of it as or refer to it as a functor, because that would make us seem very pretentious.
Best Answer
Over $\mathbb{C}$, elliptic curves with, say, a point of order $N$ can be identified with the quotient of the upper half plane $\mathbb{H}$ by $\Gamma_1(N)$ just by associating to the class of $\tau\in \mathbb{H}$ the isomorphism class of the air $(\mathbb{C}/(\mathbb{Z} \oplus \mathbb{Z}\tau), 1/N)$. Under this identification, modular forms of level $N$ can be realized as sections of a certain line bundle on "the space" (using the term loosely) of such isomorphism classes that is natural in a sense because is has only to do with this moduli interpretation (roughly speaking, the fiber at each point is a tensor power of the space of differentials on the associated elliptic curve).
One can take this observation a lot further to get a good notion of modular forms over base rings other than $\mathbb{C}$ by studying sections of these natural invertible sheaves on modular curves over these more general bases. In particular, one gets a good notion of $p$-adic analytic modular forms by looking at rigid-analytic moduli spaces of elliptic curves.
ADDED IN EDIT:
Regarding your second question, here is perhaps one reason related to my answer above to "believe" that such a moduli interpretation shouldn't exist. I'm not sure how convincing it is...
If there were such an interpretation, then one should be able to mimic the stuff in my answer above to get a geometric notion of modular forms for non-congruence subgroups over a more general class of rings, including, say, $\mathbb{Z}$. Then basic facts from algebraic geometry would kick in and tell you that the Fourier coefficients of such forms should have bounded denominators, as they do for congruence subgroups. This, however, is false for modular forms on non-congruence subgroups. I'm not sure of the history behind these results, but I know that Winnie Li and her collaborators have proven theorems in this area.