I really cannot help you much. What I want to add is that I have (experimental) Moodle's installed on three different hosted server systems, and they all work. However, I did have to make sure that the basic requirements of Moodle were met, meaning up to date php and mySQL, and all of the required php directives turned on. Having said this, there are still many ways to install, and versions of all of these php modules that when Moodle doesn't behave, it is not real surprising to me. I am not a system administrator, but somehow manage to get my moodle's working.
Ken's advice is always great, and I respect his knowledge. But I have never had to do any of the things that he has suggested to you in my own installs.
Here's a few (simple) ideas for you. Have you set the database collation to utf8_unicode_ci?
Consider comparing the results of php.info on your system to that of other successful installs. Look for differences, especially required modules and php settings. This will take awhile.
My "production moodle" system is as follows: Moodle 2.6.3 (Build: 20140512), PHP Version 5.5.7, MySQL
5.5.34, CENTOS 6.4, Apache 2.2.26. This is a GoDaddy "Value" VPS. I have around 6 experimental installs also working. Right now, on one of my installs I get a "Redirect Loop" because I just converted from a raw IP address to a domain name, but my server does the redirection just fine. I am going to fix this by editing my config.php file, redoing the URL on the few lines. Prior to assigning a domain name to the IP, this install of moodle was working fine.
Hope some of this helps.