Both examples are grammatical. In ① and ②, both either and too are working as adverbs.
Adverbs (and more generally adverbial phrases) can complement verbs, adjectives or, as in ① and ②, clauses.
I am unable to hear either.
"Either" in the above sentence complements the whole clause "I am unable to hear". It is used to indicate that the clause is adding some information to the preceeding sentence. As noted by the OP, "either" is used for this purpose only when both sentences express negation. In ①, "unable" provides this negative meaning in both sentences.
Other examples of this usage of either (taken from the dictionary linked above):
You don’t like him, do you? I don’t, either.
It won’t do any harm, but won’t really help, either.
I was too tired to go. And I couldn’t have paid my way, either.
If any of the sentences, as in example ②, does not express negation, then the adverb "either" is infelicitous, and "too" should be used instead:
Note that both "either" and "too" link one clause with the preceding sentence, and hence, there must be a reason to link them.
In the case of ①, both sentences follow the same pattern: "I am unable to".
In the case of ②, more context is needed to make this reason patent, e.g.:
Yesterday was a terrible day: I felt humiliated at the party last night. I was unable to sleep well, too.
You are right that it is not gramatically correct, but certainly in British English it is quite commonly used. It is a double negative, which if we would take literally would mean a positive. But in fact we don't take it literally, and the meaning which it still conveys is
We don't need any education
I would argue that the reason it is used in the Pink Floyd song, is for irony. This is because using no instead of any in a sentence like this, is considered to be something someone uneducated would do. You certainly wouldn't hear the Queen saying it like this!
Also, maybe the single-syllabled "no" just works better for the rhythm than "any".
Not obeying the rules of grammar, in some circles, makes one appear outrageously cool and hip to one's peers. In civilised England however, we recognise that speaking and writing grammatically is more important than anything else in the world.
Best Answer
In standard English (unlike much dialect and slang usage), "I don't see nothing" is incorrect - unless your intended meaning is "I see something". In standard English, a "double negative" results in one negation cancelling out the other negation, resulting in an affirmative meaning.
In standard English, the correct expressions are: