I need to create a custom Lightning component and inside of it I need to output record details page and related lists. There are standard components in App Builder for this: "Record Details" and "Related List - Single" but I can't find a way on how to add it to my component. How can I do this?
No you cannot include standard lighting components in the custom components as of today unless you find a documentation from salesforce that says you can do so.
Your best alternative approach to make things easy on the development part would be leveraging Lightning Data Service in your custom components whenever and wherever possible!