Which strings are in this set?

I'll show how to prove $v_1 = v_1 \in \kappa$, the other statment $\lnot (0 = 2 \land 1 = 2) \not \in \kappa$ can be shown using Godel's completeness theorem. Let $A \to B$ be a shorthand for $\lnot (A \land \lnot B)$.

Exercise 1. If $A \in \kappa$, then $\lnot A \to B \in \kappa$ for each formula $B$.

Conclude that $\lnot (v_0 = v_0) \to \lnot (A \to A) \in \kappa$ for some formula $A$. By your last rule conclude that $\exists v_0 \lnot (v_0 = v_0) \to \lnot (A \to A) \in \kappa$.

Exercise 2. Prove that $\lnot \exists v_0 \lnot (v_0 = v_0) \in \kappa$.

From the second to last rule conclude $\lnot (\lnot (v_1 = v_1) \land \lnot \exists v_0 \lnot (v_0 = v_0)) \in \kappa$.

Exercise 3. Prove that $\lnot (\lnot \exists v_0 \lnot (v_0 = v_0) \land \lnot(v_1 = v_1)) \in \kappa$.

$v_1 = v_1 \in \kappa$ by modus ponens.