The songs files for music that is on the iPod are located on the iPod. While it is connected to iTunes, you can access them using iTunes, and play them (if the iPod is set to Manually manage music). If you disconnect the iPod, the songs files are no longer accessible by iTunes. So, what you describe sounds normal.
You cannot use iTunes to transfer song files from iPod to computer, except for songs purchased (by you) from the iTunes Store. However, there are third-party methods and utilities that can transfer from iPod to computer. If you do an Internet search on something like "ipod transfer," you should get some links. (The CNET article seems to be popular.)