Unrecognizability of $L(M_1) \cap L(M_2) = \emptyset$
Let $T$ be a Turing machine. Construct a Turing machine $T'$ that simulates $T$ on the empty input word and, when (if) the simulation ends, it accepts (regardless of the input to $T'$). Notice that $T'$ can be computed from $T$.
Pick $M_2$ as the Turing machine that immediately halts and accepts.
Then $\{T', M_2\} \in C$ iff $L(T') = \emptyset$, which happens iff $T$ does not halt on the empty input word. Therefore if $C$ were recognizable, the halting problem would be decidable (since the language of all Turing machines that halt on the empty input word is recognizable).
This shows that $C$ is not recognizable.