Mmm. I checked your website, but I don't see what you described in your first message.
You first said:
If I click from the homepage into the cart, the URL to the checkout (from cart view) looks like this:
www.domain.com/checkout/chechout/
- it doubles the "checkout".
On your homepage, if I look at the "Zur Kasse" button of the cart module, the link is /warenkorb/checkout
Then, you said:
If I click from a category or product page into the cart, the URL to the checkout (from cart view) looks like this:
www.domain.com/product/chechout/
- every following step contains the product name in the url.
On your website, if I go to the menu item "Figuren > Waldifiguren" I get the URL /figuren/waldfiguren
There, the URL of the "Zur Kasse" button of the cart module is still /warenkorb/checkout
And if I open the page of a product, like /figuren/geschnitzter-holzfaeller-serie-7cm , the URL of the "Zur Kasse" button of the cart module is still /warenkorb/checkout
So the URL to access the checkout is consistent everywhere I could check on your website and it doesn't double "checkout".
So, I'm not sure what to tell you. Maybe I didn't understand what your problem is. If so, could you please try to explain again what the problem is ?