You are free to delete any files that you added to your site. Just be careful not to delete files provided by LP (including, but not limited to, configuration, mail, and web use statistics). If you installed scripts via cPanel > Fantastico that you no longer wish to use, try to use the provided "uninstall" or "remove" functions rather than deleting by hand.

You can completely restructure a website, but keep one thing in mind: if any visitors have bookmarked pages, or search engines have cataloged pages, they might be upset if you change all the names and directories. Consider using .htaccess entries to "translate" (with code 301 permanent redirect) from the old structure and names to the new (e.g., public_html/index.html is now public_html/index.php). This may be simple or in may be horrendous, depending on how you changed the site layout.