Does "за" need to be followed by a particular grammatical case? If not, how do you explain this phrase from a poem:
за твоею спиной умолкает в кустах беготня
Or is it being used for some other purpose?
Prepositions almost always require some grammatical case. There's a distinct set of cases for each one. So there may be several combinations of preposition + case, and, additionally, every pair can have more than one semantic meaning. This relates to the valency and transitivity of a verb. So in terms of за, at least two variants exist (you can check out the others in ru.wiktionary):
I just looked at wiktionary. Huh, there are a lot of meanings - 25 at least. I think some are recurrent. So it's hard to even list them here. But most are used with the accusative and instrumental.
The poem in question uses the first variant with a poetic form of the instrumental.
The preposition за takes the instrumental case, as in your example, when there's no movement involved:
Дети играют за школой. The children are playing behind the school.
However, when there's movement towards an object or destination, it takes the accusative:
Дети бегут за школу. The children are running behind the school.
In this example, the children are running towards the back of the school.
За means "behind", so in the poem, it means "there's running (as a noun this means process) going quiet behind your back".
За as behind: за спиной, за диваном. But за can be used in another way. For example: Я пошла за молокom (instrumental). I'm going/I went/I've gone to buy milk. Literally, it's more like "for milk".
Here are some more examples:
a) Сходи за хлебом! (Go buy milk!) b) Иду за джинсами (I'm going to get some jeans)
It usually works with the verbs "go" and "run", but you can use за with the meaning "to follow". To follow a bus - бежать за автобусом (or maybe to run after a bus). It looks the same, but has a different meaning.
I'm following you - я бегу за тобой, я иду за тобой.
I hope you got this. Let me know if you have any questions <333
No. As an example,
За твою спину в кусты улетел воланчик.