I am no expert, but here is what I understand about classifying CFTs in two dimensions (so I will only try to answer your first question, I do not know enough to tackle the others):
The defining property of a CFT is that the primary fields are invariant under the transformations generated by the Virasoro algebra $\mathrm{Vir}_c$ (or rather the sum of the algebra with is conjugate, $\mathrm{Vir}_c \oplus \bar{\mathrm{Vir}}_c$) spanned by $L_n, n \in \mathbb{Z}$ and a central charge operator $C$ with the commutation relations $[L_n,L_m] = (m-n)L_{m+n} + \frac{C}{12}(m^3 - m) \delta_{m+n,0}$ and $[C,L_n]=0$. Since $C$ is central, it is a Casimir operator, hence it has only one eigenvalue, whith is denoted $c \in \mathbb{R}$ , the central charge. The classification of CFTs is now achieved by understanding the representation theory of the Virasoro algebra, since, for the algebra to act upon the fields, they must transform as elements of some representation of $\mathrm{Vir}_c$.
A crucial property of useful QFTs is that they must be unitary, i.e. the space of states (which, by the state-field correspondence of CFTs, is in bijection to the space of fields) must possess a positive-definite inner product and $L_n^\dagger = L_{-n}$ must hold on them. So, we seek unitary representations of the Virasoro algebra. In addition, it is enough (and desirable) to know all irreducible representations, i.e those that do not possess a true subspace that is also a representation - meaning that this subspace is closed with respect to the action of the $L_n$ on it.
One then begins to build representations out of the so-called Verma modules $V(h,c)$. To get these, first define a vector $|h\rangle$ to be a highest weight vector, i.e. $L_0|h\rangle = h|h\rangle$ and $L_n|h\rangle = 0\ \ \forall n > 0$, and then let these modules be the span of $\{L_{n_1}\dots L_{n_j}|h\rangle | n_i\leq -1 \ \forall i\}$. This is a representation of $\mathrm{Vir}_c$ by construction, but in general neither unitary nor irreducible.
We can define an inner product on these modules by setting $\langle h | h \rangle = 0$ and extending this to all other states through the commutation relations and the unitarity condition $L_n^\dagger = L_{-n}$ - for any given Verma element $|v\rangle = L_{n_1}\dots L_{n_j}|h\rangle$, we will eventually arrive at some number if we stubbornly evaluate $\langle v | v \rangle$. For example, with $|v\rangle = L_{-1}|h\rangle$, one finds $\langle v | v \rangle = 2h$ (try it!). This is obviously negative for $h < 0$, thus this will not be a unitary representation. For $h = 0$, it is zero, and we can (in the sense that vectors with zero norm are unphysical, nicely corresponding to quantization of Yang-Mills theories) simply factor these out - $|v\rangle$ generates the subrepresentation $V(-1,c) \subset V(0,c)$. Indeed $V(0,c) / V(-1,c)$ will, for suitable values of $c$ be an irreducible representation.
One now considers these unitarity condition level by level - so we look at the norm of $L_{-2}|h\rangle$, $L^2_{-1}|h\rangle$, $L_{-3}|h\rangle$ and so on. This can actually be done in closed form (by something called the Kac determinant formula), and with a quite complicated analysis one finally concludes that:
- There are no unitary CFTs for a highest weight $h < 0$ or for a central charge $c < 0$
- For $c > 1, h \geq 0$, all Verma modules are unitary and irreducible
- For $c = 1, h \geq 0$, all Verma modules are unitary, but reducible if $h = \frac{n^2}{4}, n \in \mathbb{N}$
- For $c \in (0,1)$, only those with $c = 1 - \frac{6}{m(m+1)}, m \in \mathbb{N}_{\geq2}$ and $h = \frac{((m+1)r - ms)^2-1}{4m(m+1)}, r \in {1,2,\dots,m-1}, s\in {1,2,\dots,r}$ are unitary and irreducible. If $m = \frac{p}{p-q}$, then $r < p$ and $s < q$ produces unitary, but in general reducible representations.
So, this contains the answer to your first question: Specifying the central charge (i.e. setting a definite Virasoro algebra as symmetry) alone is, in general, not enough, one also needs to specify which conformal weights will occur. But, as there is no natural reason to prefer one weight over another, all possible weights will occur. Therefore, the representations with $c\in(0,1)$ are particularly interesting, since the allowed weights are already heavily constrained by the Virasoro symmetry. They are called minimal models.
For example, $c = 1/2$ has three distinct representations with $ h = 0,\frac{1}{2},\frac{1}{16} $, and their direct sum is where the theory of a Majorana fermion CFT takes place. This is also the theory of the continuum limit of the Ising model.
Hopefully, this is a somewhat satisfying answer to your question what lies at the heart of defining a 2D CFT.
