Display categories in views to get chosen and to display specific tags I wonder if there is a possibility to display categories and after clicking them, views display only specific category's items. Is it possible?
Yes, You can able to list taxonomy terms of specific vocabulary using views.

You need to create a custom block and a view for this..


A custom block, Which lists all vocabularies .. Views doesn't support vocabulary listing..
Two, A view which lists specific vocabulary terms. You need to pass vocabulary id to this view in an URL and read it using contextual filters to filter the terms by category.


You can get list of vocabularies using below api function..

taxonomy_get_vocabularies

Link the vocabularies terms to some path like category/list-terms/{vid} .. make category/list-terms/% as your view path and read the argument using contextual filter vocabulary id to list terms..