Thanks for your help WPTechGirl !!!
Point 1 worked as you said, posts on main site look like they’re supposed to.
So for now:
3) Slider with ‘featured’ posts still doesn’t work. ‘Featured’ tag addeed to post does nothing.
4) How can I manage the content in the black box just below featured image on single post view? I want there to be posts chosen by me – for example only with specific tag. As far as I can see there are all previous post right now.
5) I see some ‘water marks’ on featured images on the main site. On demo there are a few kinds of them (a pin, photo, video, text) – how can I choose one of them or just delete it when I want to?
Konrad