Hi Trevor,
As Jason said, the existence of both python installations is the most likely cause of the issue. What you could try doing before attempting a reinstall of ArcGIS is set some python directories in your PATH to see if that solves the problem
To set your PATH variable in Windows, go to Start >> Control Panel >> System, click Advanced System Settings, click the Environment Variables button in the lower part of the Advanced tab.
In the System variables section in the lower part of the form, scroll through until you find the Path variable.
Make sure that there are no values pointing to the C:\Python27 directory, if so delete those parts (between the semi colons). Add the following to the end of your Path variable -
C:\Python27\ArcGIS10.2\;C:\Python27\ArcGIS10.2\Scripts\
This should be enough to allow calls to python to use the ArcGIS embedded python instance by default.
I notice one exception you are seeing is coming from the PyWin module. It may have also been affected by the two python installations. Try installing PyWin32 again, and make sure during the installation that it finds the ArcGIS embedded python environment to install into.
Cheers,
Todd