Does Craft CMS support these features?
Preview function for different devices, screen sizes and screen
  configurations(portrait/landscape)


Yes it has live preview. However this isn't per device. You can open the entry with a secret url and then user your browser tools to check different devices


  Is it possible to publish only a subset of my website?


Yes, each entry can be turned on or off


  Is lazy loading supported? E.g. loading images on scrollto? Does a
  frontend created with Craft CMS use much .js code? Can I minimize the
  amount of .js code (for performance reasons)? Is mobile first approach
  supported? Are URLs SEO-friendly?


Yes, it's all how you code it. You can do whatever you want. Craft makes no assumptions or requirements


  Can I integrate Onsite-Banners from external Ad servers?


Yes code as you want


  Does the cms contain Digital Asset Management? Which media file types
  are supported?


Yes, full list here: 


  Is there any REST API provided? Does the CMS internal search engine
  support synonyms and how about the error tolerance?


I'm not sure


  Can I login to the CMS on mobile devices? Do I have good usability
  when doing so?


Yes you do. I've done this myself and it was easy to move around


  Do I upload high resolution images and later on these are generated
  and maybe compressed for target devices?


Yes you can and then you can set transforms to crop/resize/compress as you want. Targetting per device that all depends on your html.