Learn English – Preposition selection for “Are you doing anything special ….. New Year?”

Your intuition is right. The customary preposition there is “at”:

I am learning the ropes at my new job.

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The reason for “at” is to indicate a location without regard to its shape or structure. If you imagine the job as a location, there are metaphorical “ropes” there which you have to learn. At other locations, “the ropes” are different.

It’s natural to say in or of when referring to a subject or a skill:

Very few people will help you learn the ropes in science.

Learning the ropes of science can take many years.

A job naturally has a location, but science doesn’t, so “at” would sound wrong there.

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You can say this:

I am learning the ropes of my new job.

but it suggests that by “new job” you mean the occupation in general, not the specific place where you work. For example, you might learn the ropes of real estate by learning the ropes at Century 21. It's not wrong to say that you’re learning the ropes of Century 21, though.

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.