I would like to make Popular tab show the most popular posts in the last week. And I know how to do that using data I have from WordPress Popular Posts plugin. I just don’t want to change files inside the theme, but to put the changes in my child theme. Is there a way to make it work like the way I can change how a single post looks like (I mean copying single.php to the child theme folder and then editing it)?
What I get from your explanation is that you want to create widget for popular posts and making widgets have nothing to do with a theme. Widgets are part of WP core and works irrespective of the theme.
If you know how to code the widget for popular posts then make a widget out of it and add it to the sidebar you want to display.