This issue begin after i installed the Qivva calendar Joomla module. The AJAX request return ajax failure it returns 500 error. It since the ajax request of Qivva isn't towards the cause of the website, It is going for this URL

/modules/mod_eventlistcalqajax/mod_ajaxloader.php?modid=42&Itemid=4&el_mcal_month=1&el_mcal_year=2012

While, in arrears joomla chmod the modules folder to 777. my Real question is, Why php can not be performed inside chmoded 777 folder?

The code 500 means internal server error. Which means you can examine your error.log to learn more. I don't think that it's associated with the 777 permissions.