Right-click on folder in file manager and select "Edit Access Rights" or click "Change Access Rights" button in file manager toolbar to setup access rights for the folder.

You can also look at file manager in Customer zone of Alloy Tech templates if you need something like that on your site.

Does this mean that I can not give access and show a folder with its files to an end-user. I was looking for a way to present a folder and its content to the user in the same way as an Editor can see it.

You can. Set Read access rights for Everyone role for folder that should be accessible by anonymous users.

The "File Management" button show up only in Edit mode not in User mode. Is there a way to show a folder "out of the box" to a user or do I have to write some code?

All presentation to user is made through the page templates created by developers yes. So if you want to show "related files" or similar to end-user you will need a page template that has been built with that functionality. There is an example on the Alloy Tech site how to make both file listings and a simple "File Management" for a normal user. But yes, if you don't have that functionality built already on your site, you will need a developer to create that functionality for you.

Exactly. Please look at file manager in Customer zone of Alloy Tech sample site if you need file-manager-like functionality.

This blog post describes how to add file manager on Relate site.