How to copy one folder to another folder recursively

You can make use of the Filesystem API, especially the copy_dir() function – source – does seem to be what you are looking for. Take a look at the docblock from the above source link for a little bit more information:

/**
* Copies a directory from one location to another via the WordPress Filesystem Abstraction.
* Assumes that WP_Filesystem() has already been called and setup.
*
* @since 2.5.0
*
* @param string $from source directory
* @param string $to destination directory
* @param array $skip_list a list of files/folders to skip copying
* @return mixed WP_Error on failure, True on success.
*/

The (very) basic usage scenario would look like this:

global $wp_filesystem;
copy_dir( 
    '/path/you/are/copying/from/', 
    '/path/you/are/copying/to/'
);

Unfortunately the Filesystem API isn’t exactly well documented. You definitely should take a look at the linked codex article. One additional (almost always) linked source of information would be the article by @Otto:

• Tutorial: Using the WP_Filesystem

Besides that there are some Q&A’s on here which might be helpful:

• Using wp_filesystem in Plugins
• Use WP_Filesystem to list files in directory
• Copy and delete a directory with WordPress functions

Note, the above are just a small sample, use the search to find more. Last but not least, to take a closer look at the source to learn and know more about it should definitely be on your agenda.