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What came first and which one was the inspiration to the second one? Or did both came continuously where Eminem cooperated with the movie creators for the purpose? Is there any article describing their cooperation in detail?

• 
  Venom on IMDb, released 1st of October 2018 in the USA


• 
  Eminem - Venom on YouTube published 5th of October 2018 worldwide

Although the song came 4 days later, it's obvious October was a target month for both. Seems they cooperated together indeed. Where did the "Venom" idea came from?

I would point you to the Wikipedia page for this song which reads:

"Venom – Music from the Motion Picture", more commonly known as simply "Venom", is a song by American rapper Eminem, written for the soundtrack of the 2018 film of the same name and featured in his album Kamikaze.

(emphasis mine)

As far as which was released first, the same page has the following to say:

On August 30, 2018, Eminem posted a 15-seconds teaser showing Venom's title with the "E" turning to the "backwards E", used as Eminem's emblem. Seven hours later, on August 31, 2018, Eminem's previously unannounced album Kamikaze was released, and "Venom" appeared as its final track. On September 21, 2018, the track was released as a separate digital single on streaming services

So I would say, rather than the movie inspiring the song or vice-versa, the producers of the film commissioned work from the artist (he was likely well paid for this - speculation on my part).

Logically, the song would have to be finished first to be included in the end credits of the film

The song was written for the film.

Eminem has released a teaser for a new song, seemingly set to be
featured on the soundtrack for the upcoming Marvel film Venom. Posted
to Instagram, the teaser shows the Venom title sequence shown in
previous trailers for the film, with the “E” eventually morphing into
Marshall Mathers’ signature backwards letter. Find that below.

source

The Venom film was inspired by a comic book character, not a song. The character was first introduced in 1963.