In answer to the question in the first paragraph, you are correct, though it is irrelevant. Yes, a model $\langle M , E \rangle$ of ZFC could have the property that its version $\varnothing^M$ of the emptyset is actually an infinite set, but what we know for certain is that $\{ b \in M : b \mathrel{E} \varnothing^M \} = \varnothing,$ since the emptyset is defined by having the property that $( \forall y ) ( y \notin \varnothing )$. This Question is really talking about these subsets of $M$ represented by elements of $M$ (as in previous questions).
In regards to your second question, this Question has (virtually) nothing to do with whether an element $a \in M$ that $M$ thinks is finite is actually an infinite set in real life. It is saying that there is no formula $\phi (x)$ in the language of set theory such that whenever $\langle M , E \rangle$ is a model of ZFC and $a$ is an element of $M$ then $ M \models \phi [ a ] \quad \Longleftrightarrow \quad \{ b \in M : b \mathrel{E} a \} \text{ is finite}$ (and by "is finite" on the right-hand-side we mean is really finite in the real world).
It appears to me that Just-Weese will be showing that a particular theory extending ZFC will be consistent where there following are new axioms:
- $c_k \neq c_\ell$ for distinct $k , \ell$;
- $c_k \in c_0$ for all $k \neq 0$; and
- $\mathrm{fin} (c_0)$.
Then a model $\langle M , E , c_0 , c_1 , \ldots \rangle$ of this theory will also be a model of ZFC, and $M \models \mathrm{fin} [ c_0 ]$ however the set $\{ b \in M : b \mathrel{E} c_0 \} \supseteq \{ c_k : k \neq 0 \}$ is infinite. (This will be similar to an idea I sketched out in a previous answer.)