If you want to create a custom page template as a landing page, you can duplicate the index.php file and make your edits to the copy. To make it available as a selectable page template in the page/post editor, just follow the instructions under "Creating Custom Page Templates for Global Use" here:https://developer.wordpress.org/themes/template-files-section/page-template-files/#creating-custom-page-templates-for-global-use
As far as adding classes to the body content, you can edit the content.php file in the theme folder to make any edits you need. The WordPress Codex
can give you information on what pieces of code will add the classes you want.