Q: iTunes store "Purchased" files do not show up on my hard drive

This isn't completely coherent so I am not sure what is wrong other than some (al?) of your tracks have broken links.

If you are browsing through iTunes you are not looking at files you are looking at data entries that point to files, with those locations being incorrect right now. 

There isn't a file called iTunes media, there is a folder that should be in your iTunes folder called that.

You mention something about external drives. I presume one is for your Time Machine backup but is the other one used for iTunes?  If so, what do you keep there?

Here's some background information to help place this in context:

What are the iTunes library files? - http://support.apple.com/kb/HT1660

More on iTunes library files and what they do - http://en.wikipedia.org/wiki/ITunes#Media_management

What are all those iTunes files? - http://www.macworld.com/article/139974/2009/04/itunes_files.html

Where are my iTunes files located? - http://support.apple.com/kb/ht1391

iTunes 9 [and later]: Understanding iTunes Media Organization - http://support.apple.com/kb/ht3847