This has been happening for about two months and it makes me so angry and annoyed since the phone calls stop the music I'm listening. to. (or [...] interrupt the music I'm listening to. / [...] pause the music I'm listening to.)

I wasn't at home when the rain started, so after a couple of minutes I could return anded to close all the windows.

(I'm not sure if I corrected this sentence to the meaning you intended)

It was a heavy rain but it didn't last for so long.

It's a bit unusual to put "a" before "rain" but it is grammatically correct if the "rain" is treated as an event. To make this more natural, you can just say "There was heavy rainfall, but..."

I'll hang out with a friend at McDonald's soonin a bit.

"Soon" here sounds vague, like it could mean "soon" as in "in a couple of days" instead of "later", so I corrected it to "in a bit".

We both are not into alcoholic drinks so we usually go out tonly for eating or playing volleyball, avoiding parties like "weird nerds".

For those who don't know what City Pop is, it's basically jJapanese 80's music.

The atmosphervibe from those songs is amazing.

It's unusual to use "atmosphere" here, so I corrected it to "vibe". "Atmosphere" seems like you're talking about a place, whereas "vibe" has a more intangible meaning.

I only had time to get myself a mug of coffee with milk which I basically drink every morning, along with some crackers or something else.

A lot of unknown phone numbers started calling me during work, - almost non-stop.

The hyphen gives you a long break to show that it’s a related but not directly following on sentence.

This has been happening for about two months and it makes me so angry and annoyed since the phone calls stop the music I'm listening. to.

I wasn't at home when the rain started so afterbut I wasn’t far away and it only took a couple of minutes I could returnfor me to get home and close all the windows.

If you say return, I feel like you just left and ran back. I would say “got home from work” and not “returned from work”.

It was a heavy rain but it didn't last for sovery long.

We both areare bith not into alcoholic drinks so we usually go out only forto eating or playing volleyball, avoiding parties like "weird nerds".

The atmosphere from those songs is amazing.

We use the word atmosphere more for places than for things. I can’t think of a way we might say this in English without it sounding weird. Maybe just “they’re great/amazing”