Terminology – Difference Between Articles and Determiners

Question 1 - She doesn't need to say "the beans", because it really doesn't matter which beans she's getting. Whether Aki gives her those beans, or some different beans, the question would still be the same. So "the" would be unnecessary. And she could say "some beans", but this would be repeating what Aki had already said. Again, unnecessary.

Question 2 - If she didn't say "the", it would imply that the question related to beans in general ("why beans but not lentils for Setsubun"); but with "the", it really just means "why did you give me those beans".

Question 3 - If the beans that get thrown were the "special holy beans of Setsubia, which are removed from the sacred vial every Setsubun", we might say "we throw the beans". But no, any beans will do; so Aki just says "we throw beans". She could also say "we throw some beans, but not too many", but because she doesn't care about the quantity, she doesn't need to say "some'.

The word "conjoiner" is sometimes used in linguistic theory, but rarely in terms of English grammar.

A conjunction in English refers to a specific set of words; "and," "or," "but," "although." A conjunction is a part of speech, like a noun or a verb.

A conjoiner is a linguistics concept that refers to any linguistic construct that plays the same role. In some languages, the role of some conjunctions (especially subordinating conjunctions) are played by units other than independent words, such as verb endings.

Neither has anything to do with articles, which are separate parts of speech.

If your sentence was:

*The river ran down to sea

you would say:

You're missing an article in that sentence.

If your sentence was:

*The river wound its way down the mountain through the forest

you would say:

You're missing a conjunction in that sentence.

You would never tell someone they were missing a conjoiner, unless you were a theoretical linguist talking to another theoretical linguist.