Superset of another language and recognizability of turing machines

$ L_1$ = $ \{\langle M \rangle \mid M$ is a turing machine and $ M$ halts on some string$ \}$

$ L_2$ = $ \{\langle M \rangle \mid M$ is a turing machine and $ M$ halts on all strings $ \}$

a) Is $ L_2$ a superset of $ L_1$ ?

b) Is $ L_2$ not co-recognizable and not recognizable ?

(without formally proving)

my attempt

$ a)$ $ L_2$ is a superset of $ L_1$ since all strings are in $ L_2$ whereas $ L_1$ is some string

$ (b)$ $ L_2$ is not recognizable because, out of the infinite enumerations of strings, a single one could not halt disproving it from being recognizable.

For not co-recognizable: It can be written like this

$ \bar{L_2} = \{ \langle M \rangle \mid M$ is a turing machine and $ M$ loops on some string $ \}$ .

It’s not possible to make a recognizer for loops so its instantly not co-recognizable. This is because looping means it never stops so to come up with a recognizer for anything to prove it loops is impossible since we are working with infinite combinations.

Not sure about it being a superset or not and my explanation for co-recognizable