I've been wondering what Enterprise Search Center has to do with (if anything) Content Search Results web part and/or Content Search web part. Do I need Enterprise Search Center for the queries to work in Content Search Results web parts?
Their is no relation between Enterprise Search Center & Search Web Parts. Only requirement for the Search Web Parts are, You have to Configure the enterprise Search Services Application.
Web Parts that use search technology to show content in a publishing environment (referred to in this article as Search Web Parts) show content that was crawled and added to the search index, These Web Parts have queries defined in them, and when users browse to a page that contains a Web Part that uses search technology, the Web Part issues the query automatically. The query result is then displayed in the Web Part. You can modify the query in the search Web Part to fit your content needs.
Please check this for more info: http://technet.microsoft.com/en-us/library/jj679900(v=office.15).aspx