Writing the core as the intersection of pareto efficient outcomes of all coalitions
Most of what you write is correct, but the definitions of the $F_i$ sets is imprecise. The problem is that in the core $A$ and $B$ may get goods that do not match their initial endowments. In this case it is not true that the core allocation $x$ is Pareto-efficient in the restricted 2-person economy of $A$ and $B$, because $x$ is not even an allocation in that game.




Edit: (An example)

Consider the initial endowments
$$
\omega_{A} = (1,1),  \omega_{B} = (1,1), \omega_{C} = (2,2)
$$
and an allocation $x$
$$
x_A = (2,2), x_B = (2,2), x_C = (0,0).
$$
$A$ and $B$ cannot Pareto-improve on $x$. But $x$ is not a Pareto-efficient allocation in their 2-person economy, because it is not a feasible allocation of their economy:
$$
x_A + x_B \neq \omega_{A} + \omega_{B}
$$



A better definition for the sets $F_i$  would be something like:

Let us denote the set of feasible allocations of the 2-person economy of $A$ and $B$ by $Y_{A,B} \subset \mathbb{R}^{2}$. Then
$$
F_{1} =\{ x \in \mathbb{R}^{2} : \nexists y \in Y_{A,B} \text{ such that } u_{A}(y_{A}) \geq u_{A}(x_{A}), u_{B}(y_{B}) &gt; u_{B}(x_{B}) \}
$$
There are still some issues with cases when $A$ is better off and $B$ is not worse off, but if the utility functions are continuous then this should not cause a problem.

You can define $F_{2}$ and $F_{3}$ in a similar manner.

A remark: $E$ is not 'special', it is the set of allocations that cannot be improved upon by the three player coalition. This is equivalent to Pareto-efficiency.


Yes. Why not? This is exactly what the core is.
I would not call it solving, because usually you do not get a unique solution, and in extreme cases you may get no solution. But yes, every economy ($n$ players, $m$ goods) has a core, and it is defined in this way. (As indicated, unless some conditions are met the core may be empty. A competitive equilibrium is always an element of the core, so if that exists, the core is non-empty.)