Hi
Today i've downloaded and installed the CS.
But if I want to launch it, only the "Steam start window" appears, which tells me, that Steam is starting the game, but nothing happens, and the "start window" disappears s few seconds later.

Don't know what is happening, but try to start Steam first then CK after it.

Hmm
should the Cs be installed in the Skyrim folder ?

You shouldn't have to worry with that:

-Install Steam

-Start Steam

-Install SKyrim

-Go to library and install the CK

-Start CK

All is automated

The problem is the CK wont start

Don't know what else to tell you, see if you meet the minimum requirements?