This suggestion depends on how your hosting provider deals with mail headers:
So, I have made all tests (javascript conflicts, etc.) and decided the problem could only be from my host. I contacted them and they told me that in email header, the "From:" SHOULD be exactly the same as the email I configured to receive the messages from my visitors.
As far as I understood, by default "Contact Form 7" uses visitor email to put it in "From:" but some host providers do not allow that.
My host provider don't even allow mail() function so I had to install WP MAIL SMTP.
So, resuming, I just added this to all my forms in "Additional headers":
From: [email protected]
This means, you have to insert one email with same domain name as your website, otherwise your hosting might not send the email.
I lost a couple of hours with this...