My issue was identical.... days of frustrating downloads and initializing but no installation........ UNTIL I chose to not use the default installer and chose 'Save File'. Now I have the latest Flash installed in OS 10.8.5 no problems !!
This time a long download of 16MB arrived in the downloads folder that actually worked.
Prior to this, all downloads were around 1.6MB and there was no .dmg in the package.
I got my download under where it says "Still having problems?"
Hope this helps you.
See the attached page for reference.