Okay, I got it. I had to copy another part of the parent css to the child css and modify it to my needs. Thanks!
I’ trying to figure out what happens using Firefox’s integrated web developer tool, but currently I’m stuck. The site isn’t finished yet, but you can find it here: bit.ly/1LEzd7g
What’s astonishing is that the classes “icon-pdf” etc. work just fine inside the div with the class “content-area”, so I added different class names to the child style.css to use it inside the footer area as well. I tried the ones mentioned in my first post and also “site-footer”, but none of the ones I tried worked.