Slideout Navigation,. GeneratePress Premium Version: 2.3

Ah, in that case, keep it set to "Both" and add this CSS:

.gen-sidebar-nav .slideout-toggle {
    display: none !important;
}

ok, that has worked out.
Unfortunately, the mobile menu is no longer sticky
- I did not test that enough yesterday so I did not notice it

And the mobile header is still set to sticky?

yes

Hmm, sticky.min.js isn't being added to your page, which means WordPress is reading that option as false. No caching plugins on the site? Cloudflare?

No, it's just the plugins
Breadcrumb NavXT
Classic Editor
Contact Form 7
Easy Updates Manager
GP Premium
Limit Login Attempts Reloaded
Self-Hosted Google Fonts
Shortcodes Ultimate
Table Press
WPML
installed.

The menu is now sticky, the hamburger icon is not.

I think it makes more sense to rebuild the menu but without knowing what the problem was / is, this is not the best way.

That's because you've moved the menu toggle into the top bar, which isn't sticky.

You could try this instead:

.top-bar {
    position: sticky;
    top: 0;
}

ok, thanks!
now I am happy :-)

I would have to clean up the server again, should I upload and link the pictures from my postings elsewhere?
Do you have a favorite hoster?

You can find some of our recommendations here: https://docs.generatepress.com/article/recommended-wordpress-hosting/