Hi Philip,
[2014-02-04 02:53:48] *** Error: Unable to remove directory: Permission denied: /Applications/Astronomy/PixInsight.app/Contents/doc/docs
Here is the problem. Your PixInsight application bundle is write-protected, or, more exactly, the user you are running as does not have permission to write or change files on the "/Applications/Astronomy/PixInsight.app" directory.
There are several ways to solve this problem. The easiest way is to grant write permission to the user on the PixInsight.app directory. Since the PixInsight application and all of its contained executable files are digitally signed, this should pose no actual risk to your system.