Module repeats in Child Pages

I assumed that a hidden menu is not supposed to be shown

Yes, it is hidden / not be visible, but creating a menu for your article category, it is actually like providing a clear item ID for your article categories.

When you set a module to only appear on the HOME page, then when an article link is clicked through the module, it opens a new page without any item ID information for the target page. Therefore, the module uses the previous ID, which is the HOME item ID, and the module is still displayed because it follows the assignment rules.
When you define a menu for a category, it means that you register an item ID for that specific category. So when an item in a module is clicked, there is information that the target page has a different item ID from the HOME page, and the module is not displayed, because you assign the module to be displayed in the HOME only.

Regarding the naming of 'hidden' menus (you can name it with 'not visible menu' or any else), it is actually more about to not displaying the menu on the website, for example, if all menu items are displayed, you probably have too many menu item on your website. You also can hide this menu by changing setting the Display in Menu rule to No, in the menu configuration page in the Link Type tab.
Why should it be like this? It all goes back to Joomla's basic behavior. This is just how it works. Joomla developers may have several considerations as to why this is done such as for better access control, etc.

Hope this explanation helps

1) without the hidden menu enable, the website returns to the home page; 2) with the hidden menu enable, it shows a different article in the language selected.

I'm not familiar with Falang, so I think you should ask the Falang developer about it, as creating a menu for a specific article category is a common and standard practice in Joomla, and Falang should work well with that.