CHMOD 0777, but WordPress still not able to write files/folders [closed]

This really isn’t a WordPress issue. That said, the key error is “open_basedir restriction in effect.” If you’re running your own VPS, you need to learn to configure it.

open_basedir restrictions keep scripts in one directory from being able to affect scripts in another directory, which is an important security feature. If one accounts gets hacked, open_basedir prevents the infection from spreading to another account.

See https://stackoverflow.com/questions/1846882/open-basedir-restriction-in-effect-file-is-not-within-the-allowed-paths and modify open_basedir configs http://www.php.net/manual/en/ini.core.php#ini.open-basedir

With open base configured, you should be able to go back to 755 permissions folder and the standard WP permissions scheme: Changing File Permissions « WordPress Codex