Welcome to autodesk discussion group. Thanks for reporting this issue.
From the log file, I can see the installation error 1603 when install AutoCAD Mechanical 2013.This typically related to the user right, but there are some reasons can cause this type of error. So, the following is my suggestions:
1 Please try to install with another admin user, recommend the system default user "Administrator".
2 Please turn off any antivirus and firewall, including the related processes on the background.For such applications may affect installaion files register, and may slow down your installation.
3 Please turn off UAC
4 If all above not work, please try to copy all the installation files to your local and install from there.