Joomla 3.4 folder permission issue - php

Question

I have migrated my site to new server using Akeeba Backup.
The environment of new server is Ubuntu 16.04 LTS + nginx 1.10.0 + Joomla 3.4.8 + PHP 5.6
I can see my site running well and I can also login to admin tool, which means nginx, php-fpm, and mysql are running well.
However all files/folders in System Information -> Folder Permission are all Unwritable.
I have tried any possible solutions I can figure out, can anyone help?
php-fpm and nginx is run by www-data.
$ ps aux|grep fpm
root 1293 0.0 3.4 249700 20592 ? Ss 06:27 0:00 php-fpm: master process (/etc/php/5.6/fpm/php-fpm.conf)
www-data 3397 0.2 7.5 259964 45384 ? S 06:57 0:04 php-fpm: pool www
www-data 3403 0.2 7.5 259952 45244 ? S 06:57 0:04 php-fpm: pool www
www-data 3405 0.2 9.0 269948 54716 ? S 06:57 0:03 php-fpm: pool www
$ ps aux|grep nginx
root 1348 0.0 0.2 125192 1500 ? Ss 06:27 0:00 nginx: master process /usr/sbin/nginx -g daemon on; master_process on;
www-data 1349 0.0 1.1 125584 6840 ? S 06:27 0:00 nginx: worker process
All files are 644 and folder are 775 under /usr/share/nginx/html
-rw-r--r-- 1 www-data www-data 15 May 22 15:18 404.html
-rw-r--r-- 1 www-data www-data 15 May 22 15:18 50x.html
drwxrwxr-x 10 www-data www-data 4096 May 22 15:18 administrator/
drwxrwxr-x 2 www-data www-data 4096 May 22 15:18 bin/
drwxrwxr-x 4 www-data www-data 4096 May 22 15:18 cache/
drwxrwxr-x 2 www-data www-data 4096 May 22 15:18 cli/
drwxrwxr-x 33 www-data www-data 4096 May 22 15:18 components/
-rw-rw-r-- 1 www-data www-data 3240 May 24 07:18 configuration.php
-rw-r--r-- 1 www-data www-data 586 May 24 06:59 dir.php
-rw-r--r-- 1 www-data www-data 2915 May 22 15:18 htaccess.txt
drwxrwxr-x 6 www-data www-data 4096 May 22 15:18 images/
drwxrwxr-x 2 www-data www-data 4096 May 22 15:18 includes/
-rw-r--r-- 1 www-data www-data 1211 May 22 15:18 index.php
-rw-r--r-- 1 www-data www-data 20 May 22 15:18 info.php
-rw-r--r-- 1 www-data www-data 1873 May 22 15:18 joomla.xml
drwxrwxr-x 6 www-data www-data 4096 May 22 15:18 language/
drwxrwxr-x 5 www-data www-data 4096 May 22 15:18 layouts/
drwxrwxr-x 15 www-data www-data 4096 May 22 15:18 libraries/
-rw-r--r-- 1 www-data www-data 18092 May 22 15:18 LICENSE.txt
drwxrwxrwt 2 www-data www-data 4096 May 23 14:46 logs/
drwxrwxr-x 32 www-data www-data 4096 May 22 15:18 media/
drwxrwxr-x 36 www-data www-data 4096 May 22 15:18 modules/
drwxrwxr-x 29 www-data www-data 4096 May 22 15:18 plugins/
-rw-r--r-- 1 www-data www-data 53 May 22 15:18 README.md
-rw-r--r-- 1 www-data www-data 4213 May 22 15:18 README.txt
-rw-r--r-- 1 www-data www-data 296 May 22 15:18 robots.txt
-rw-r--r-- 1 www-data www-data 842 May 22 15:18 robots.txt.dist
drwxrwxr-x 9 www-data www-data 4096 May 22 15:18 templates/
drwxrwxrwt 2 www-data www-data 4096 May 22 15:18 tmp/
-rw-r--r-- 1 www-data www-data 1690 May 22 15:18 web.config.txt
I also write PHP script to check the permission and it shows files under /usr/share/nginx/html/ are still not writable.
<?php
$root = getcwd();
echo exec('whoami')." ";
echo "$root ";
echo is_readable("/usr/share/nginx/html") ? 'yes ' : 'no ';
echo is_writable("/usr/share/nginx/html") ? 'yes ' : 'no ';
echo is_writable("/tmp") ? 'yes ' : 'no ';
?>
The result shows "www-data /usr/share/nginx/html yes no yes" even the owner is www-data:www-data and mode is 755 on /usr/share/nginx/html

Answer 1

If you create a test php file in your root directory with just
<?php echo exec('whoami'); ?>
and then access it with a browser, do you see www-data or is it something else?