If I understand your needs correctly, you would like to have your /blog page and post items aggregated and driven by WordPress without exposing the user to anything WordPress.
To solve the problem with “wordpress” being in the url, you should actually be putting the entire contents of the WordPress install into the website.tld/blog directory. Make sure you update any custom code you wrote to match the new location.
The next step would be to use style information from the current look of your website on the active theme’s style.css, modifying it to change the look of the generated lists and pages.
I don’t think a single.php can be used anywhere outside of a theme directory.