Hi, I see you've requested proofreading for this song so I've included my recommendations below. Feel free to let me know if you have any questions about any of my suggestions. Many of them are in regards to the tone of the song, hip-hop/rap should sound very informal and colloquial.

Title: No-Name (needs a hyphen)

1. Today I will break records, back then (a) No-Name -> Today I break records, back then I was a no-name
-You need the article here, so just add it without putting it in parentheses

2. Previously cheap-watches, today I wear (a) Roli -> Used to wear cheap watches, today I wear a Roli
-Same as #1

3. I said to my manager, we could go gold -> I told my manager we could go gold
-I don't think you need the footnote about going gold, it's the same in English.

4. Taman, everything is okay, yes, we start shooting, ey -> Taman, everything's okay, yeah, we start shooting, ey
-Use contractions. It sounds very formal and robotic without them, so it doesn't fit the tone of hip-hop/rap.
-Use "yeah" instead of "yes" for the same reason as above

5. Wouh! On my Iphone -> Woo! On my iPhone

6. I write a new song once again -> I write a new song again
-"Once again" is also a bit too formal for hip-hop/rap

7. It will be a lousy earwig once again -> It'll be a lousy earworm again
-Same as #4 
-Same as #6
- An earworm is a song that gets stuck in your head, an earwig is a real insect.

8. That's too bad, my dear! -> That's too bad, my man!
-"My dear" is too old-fashioned for hip-hop/rap. Use something else like: my friend, sweetie, darlin'...

9. I'm (a) Champions League -> I'm in the Champion League -OR- I'm Champion League

10. While I'm doing (this) now -> While I'm doing this now
-There's no grammatical reason to use parentheses here

11. They are laughing themself silly, I know, They are right -> They're laughing till they piss, I know, they're right - OR - They piss themselves laughing, I know, they're right
-"Silly" sounds a bit childish for this context
-Same as #4 - contractions

12. Because this song will bring reggae along in trend for me -> 'Cause this song'll bring reggae into style for me

13. Never faked again -> Never faked anymore

14. I wear the Givenchy-jacket with pelt -> I wear the Givenchy jacket with fur

15. Yes, produced just for me -> Yeah, made especially for me

16. When I go, I come back -> When I go, I'll come back - OR - If I go, I'll come back

17. Digga, what even is a Benz? -> Digga, what's a Benz, anyway?

18. A hit again, I'm fit as always -> Another hit, I'm always fit

19. Digger Buscher, certainly dangerous.-> Digger Buscher, dangerous for sure

20. I'm honest to you, rhyme gloriously, you see, too bad -> I mean it, rhyme gloriously, you see, too bad

21. Know streets, driving in a black ferrari - Know the streets, driving in a black Ferrari

22. night life fucks my brain, Bre -> Nightlife fucks my brain, Bre