Why it’s wrong
Yes, it’s grammatically wrong. But your way of thinking about English grammar is right in a very profound way: wondering if singular and plural could be exploited to suggest some subtle distinction.
Here’s why it’s wrong. The pronoun their calls for a plural antecedent. Hair is singular. Also, movement is singular, so it has to be grammatically tied to an individual hair. If you tie it to the mass of hairs all together, then it suggests that the mass of hairs all together (what we normally call “hair” with no determiner) has one movement as a whole.
How to do it right
There are ways to use singular vs. plural to indicate what you have in mind. Here’s how you’d do it:
… each of his brown wavy hairs had a movement of its own…
The singular word each gives the singular movement something to attach to, and hairs is plural. This makes it clear that the sentence is talking about many movements, not just one.
You could also indicate separate movement of each strand of hair like this:
…his brown wavy hairs had movements of their own…
You are right that logic often trumps rigid grammar rules, leading a reader to interpret a sentence reasonably when too-strict application of grammatical regularity would lead to clumsiness. But since the language provides a straightforward way to indicate the intended meaning in this case, there’s no pressure to bend grammar.
The inevitable complexity
Their can take a singular antecedent when it stands for a person and you’re trying to avoid indicating “their” gender. However, to many people's ears, this usage sounds sloppy or ungrammatical, or at best informal, because their calls for a plural antecedent. There is currently something of a war going on in the language right now, to allow their to refer to a singular person as antecedent in order to avoid sexist language. Perhaps after that war is won, their will broaden to allow singular antecedents of all kinds, but today such a development is beyond the horizon.
We can rephrase the sentence slightly, with emphasis:
The range of purposes that gongs are used for includes as a military signal, [as] a rhythmic accompaniment, and [as] a ritual instrument.
With this in mind, most of your questions become hard to parse, because include as isn't a coherent unit. I would just point out that you should not replace including with which include/s. This isn't because you can't do so, grammatically, but because it's not clear whether the verb should agree with (singular) range or (plural) purposes, and whichever you choose, nitpicking pedants will come out of the woodwork to correct you.
Best Answer
Regard is tricky. The ‘rules’ below are those observed in formal writing; you may safely ignore them in conversation.
When used as a noun to express your attention to a particular topic, it should be used in the singular:
This is often confused, even (or perhaps particularly) by native speakers with the related phrase as regards:
This is not a plural, however, but a 3d person singular verb; the sense is that your discourse now regards (“looks back to”, “contemplates”) the noun or NP which follows. The verb sense is also used in the present participle regarding:
The plural noun regards is used only in the sense “expression of esteem or affection”. In this sense the plural varies freely with the singular.
In the sense of “esteem, honor, opinion”, however, only the singular is ordinarily employed.
Respect, which also acts as both a noun and a verb, works exactly the same way in these senses.
Both words have other senses, however, in which their use varies.