Superset of another language and recognizability of turing machines
If a machine halts on all strings then it halts on some string. Therefore $L_2 \subseteq L_1$.

If a machine halts on some string but not on all strings then it is in $L_1 \setminus L_2$. Since there are such machines, $L_1 \subsetneq L_2$.



The language $L_2$, usual known as TOT, is well known to be $\Pi_2$-complete. In particular, it is neither r.e. nor co-r.e. This can also be proved directly by reducing both the halting problem and its negation to $L_2$:


Given a Turing machine $M$ and an input $x$, construct a Turing machine $M'$ which erases its input and then runs $M$ on input $x$. Then $M$ halts on $x$ iff $M' \in L_2$.
Given a Turing machine $M$ and an input $x$, construct a Turing machine $M'$ which on input $n$ simulates $M$ on $x$ for $n$ steps, and loops if $M$ halted within $n$ steps. Then $M$ doesn't halt on $x$ iff $M' \in L_2$.