The sidebar on my website includes the login/register-modul, the latest articles and random articles.
This sidebar should be shown everywhere but beside the kunema forum (see other topic) and the register modul links.

The problem now is, that if I click links of the random articles there is no sidebar beside of the articles. it seems, this happens only if I click links from the homepage.
If I click random articles on any other page everytging seems to be ok.
I can`t find the solution because I don`t understand the problem.

Then look at the register link. It has changed somehow, maybe because of the joomla update.
It was 100% width, now it isn`t.

This is related to your menu structure.
Make sure you have mapped your Joomla! categories into menu items.

Then go to module manager and recheck each module's params. Make sure that these modules are set to appear on these menus.

An other approach would be to use the "All except those selected" option.

How do you mean that?
Do I have to create a menu item for every single article I will create?
Then I don`t need the random article block, because there are already menu items...

Why this happens only from homepage?
The menu on all other pages is the same, but there it works.
For me this is not logical...

No, but it is considered a good practice to map categories (not articles) in menu items.

If you have done this already, make sure your modules are published in these menu items.

Because of Url-rewrite the categories can`t be linked in a separate menu.
It ends in 404.
However, I deactivated the random articles at the homepage.
On all other pages it works, although there is still no link in a menu to categories or articles.

I cannot understand, why there is the difference between homepage-random articles and other-pages-randomarticles links.
Usually it has to be the same result, which is shown in sidebar menus.

Because of Url-rewrite the categories can`t be linked in a separate menu.
It ends in 404.

I think you need to read this: docs.joomla.org/How_to_control_module_display_when_linking_to_an_article_with_no_menu_item

I cannot understand, why there is the difference between homepage-random articles and other-pages-randomarticles links.

This is because if no there is no menu item and the articles are rendered through the current menu. The same applies if there is a menu item with no published modules.

I solved the problem much easier, now.
I simply copied the latest news modul and changed options to random articles.
Then I can leave everything like it is and it works...