I have site at v-Boge.org - all articles are K2. Found quite a wierd thing: When I make one menu k2-category, that menu item will show me category image and description and under it articles of that category. But when I create exactly the same menu for another category - it won't give me the category image and description. I've checked all settings to be the same both for both menus and for both categories. Even compared sql lines for both menus and categoris - can't trace what makes the difference! Anybody could advise?

PS I even rebuilt the whole site from scratch thinking it might be because the site is old and accumulated bugs - to no avail... Really need a helping hand...

Check your site with another template to eliminate to possibility of a template error.
Now if your menu item points to one category the category's settings will be respected. On the other hand if your menu item is towards multiple then the menu's settings will be respected.

One last thing to check is if that category is inheriting its settings from another category.