I wonder if there is an option in K2 to allow any visitor of my page to add an article without registration? For security reasons there also should be a recaptcha for the submitting form enabled. So the article is ready to be published by an admin in backend.
Can anyone help me, please? I don't know which files I should edit to achieve this.
No, there is no way for Guests or Public visitors to post anything in neither K2 nor Joomla itself.
They have to belong at least to Author group, which is lowest group with front-end editing rights, and to be logged-in on the front-end.
That is by design. You might want to check official documentation Access Control List Tutorial.