Website is not rendering CSS and designs

In our Dispatcher, designs for the website pages are present in the path /etc/docroot/mysite. Whenever we are open any page, in the console it is not loading the path of the designs (/etc/docroot/mysite). We found that the pages loads only the content but not the Css and designs. Please find the attached screenshot for one which is working fine and the other not working. We have checked the dispatcher settings as well by adding the rule to allow folders under /etc but it didnt worked out. Please provide your suggestions.

Similarly, in our Publish Instance the above mentioned issue is occuring whenever we tried to load any page without logging into the Publish Instance. But, if we logged-in and refresh the page, everything is working fine as expected. Not sure why the external users without logging into the publish instance are facing the above mentioned issue.

Update 1: While hitting the console I am getting the following error:

 http://my-dev.website.com/etc/clientcontext/legacy/config.json?path=%2F&cq_ck=1465275433288 404 (Not Found)