Learn Hardcore Russian: Иногда я слушаю эту прямую трансляцию как фон для расслабления. - Sometimes I listen to this live broadcast as background for relaxation.

The direct object in Russian usually takes the accusative case.
эта прямая трансляция is nominative (the form used for the subject).
For a feminine noun like трансляция, the accusative singular form is трансляцию.
The demonstrative эта and the adjective прямая must agree with трансляция in gender, number and case, so they become эту прямую трансляцию (feminine, singular, accusative).

Why is it я слушаю and not я слышу? Both mean to hear / to listen, right?

They are different, like English to listen vs to hear:

слушать = to listen (intentionally)
You choose to pay attention with your ears.
Example: Я слушаю музыку. – I listen to music.
слышать = to hear (perceive sound)
You notice sounds, not necessarily on purpose.
Example: Я слышу музыку. – I hear music.

In your sentence, you choose to use the stream as background noise, so it’s an intentional action → я слушаю.

Why is слушаю (imperfective present) used here and not something like послушаю?

Слушаю is imperfective, present tense, and here it expresses a repeated / habitual action:

Иногда я слушаю… = Sometimes I (tend to / usually) listen…

If you said Иногда я послушаю эту прямую трансляцию, it would sound unusual; послушаю is perfective and normally talks about one completed action in the future:

Я послушаю эту трансляцию позже. – I will listen to this stream later (once).

So слушаю is used because you are describing what you sometimes do, not a single future event.

Can I move иногда somewhere else, like Я иногда слушаю эту прямую трансляцию…?

Yes, several word orders are possible and natural:

Иногда я слушаю эту прямую трансляцию как фон для расслабления.
Я иногда слушаю эту прямую трансляцию как фон для расслабления.
Я слушаю иногда эту прямую трансляцию как фон для расслабления. (less common, but still possible in the right context)

The meaning (“sometimes”) is the same, but the focus changes slightly:

At the beginning (Иногда я слушаю…) – neutral, very common; introduces the idea of “sometimes”.
After the subject (Я иногда слушаю…) – also very natural; a tiny bit more focus on I and what I sometimes do.

For a learner, Иногда я слушаю… and Я иногда слушаю… are the best choices.

Is it necessary to say эту? Could I just say Иногда я слушаю прямую трансляцию…?

You can drop эту, and it will still be grammatically correct:

Иногда я слушаю прямую трансляцию как фон для расслабления.

The difference is:

эту прямую трансляцию – this specific live stream (the listener knows which one you mean: a particular channel, a particular show, etc.).
прямую трансляцию – a live stream in general, not necessarily a specific one.

So use эту if you mean “this particular stream I usually listen to.”

What exactly does прямая трансляция mean? How is it different from прямой эфир or стрим?

All three are related but not identical:

прямая трансляция – live broadcast (more neutral/formal; can be TV, radio, online).
прямой эфир – literally live air; often used for live TV/radio shows:
- Мы в прямом эфире. – We are live (on air).
стрим – a modern borrowing from English stream:
- Often used for online streaming, gaming streams, Twitch, YouTube etc.
- Colloquial / internet style.

In your sentence, эту прямую трансляцию sounds neutral and is widely correct. In an informal, internet context, people might also say:

Иногда я слушаю этот стрим как фон для расслабления.

Why are эту and прямую in that form? How do they agree with трансляцию?

Russian adjectives and demonstratives must agree with the noun in:

Gender
- трансляция is feminine.
Number
- It’s singular (one stream).
Case
- It’s the direct object → accusative.

So:

Feminine, singular, nominative:
- эта прямая трансляция (this live stream – as subject)
Feminine, singular, accusative (needed here):
- эту прямую трансляцию (this live stream – as object)

All three words are “feminine, singular, accusative”, so they match.

What does как фон literally mean, and why is фон in that form?

Literally, как фон means “as background” or “as a background”.

как = as / like
фон = background (sound, image, general atmosphere)

Here, фон is in the nominative form. After как meaning “in the role of, as something”, Russian normally uses the nominative:

Он работает как учитель. – He works as a teacher.
Я использую это как пример. – I use this as an example.

So как фон = as background, in the sense of “in the role of background noise.”

What is the difference between как фон, на фоне, and под in this kind of sentence?

They have different uses:

как фон – “as background (noise)”
- Focus: you’re using something in the role of background.
- Иногда я слушаю эту прямую трансляцию как фон для расслабления.
  I use this stream as background to relax.
на фоне – “against the background of”
- Often used about images, scenery, or a situation:
- Мы сфотографировались на фоне моря. – We took a photo with the sea in the background.
- With sounds, it describes something happening while something else is in the background:
  - На фоне музыки они разговаривали. – They talked with music in the background.
под (+ accusative, with music) – “to / under (music)”
- Often used when an activity happens to the accompaniment of music/sound:
- Я работаю под музыку. – I work to music.
- Я засыпаю под шум дождя. – I fall asleep to the sound of rain.

Your sentence could be rephrased as:

Иногда я слушаю эту прямую трансляцию под расслабление. – not very natural.
More natural: Иногда я расслабляюсь под эту прямую трансляцию. – Sometimes I relax to this live stream.

But как фон emphasizes that the stream is just background noise, not the main focus of your attention.

Why is it для расслабления and not an infinitive verb like чтобы расслабиться?

Both patterns are possible, with a nuance:

для расслабления – literally “for relaxation” (uses a noun).
чтобы расслабиться – “in order to relax” (uses a verb in the infinitive).

Your sentence:

Иногда я слушаю эту прямую трансляцию как фон для расслабления.
= Sometimes I listen to this live stream as background for relaxation.

You could also say:

Иногда я слушаю эту прямую трансляцию как фон, чтобы расслабиться.
= Sometimes I listen to this live stream as background (in order) to relax.

Both are correct.
для расслабления sounds a bit more neutral/formal, чтобы расслабиться is a little more direct/personal.

Why is расслабления in that form? What case is it?

Расслабления is in the genitive case, singular.

The preposition для (“for”) almost always requires the genitive:

для отдыха – for rest
для работы – for work
для сна – for sleep
для расслабления – for relaxation

So:

Nominative: расслабление (relaxation)
Genitive after для: для расслабления (for relaxation)