Automatic "syncing" of content (such as song files) using iTunes is generally only in one direction, from computer to iPod. So, iTunes will not put songs back on the computer's hard drive from the iPod using syncing. The assumption for syncing is that everything on the iPod is already in the computer's iTunes library.
Previous song purchases from the iTunes Store can be re-downloaded at no cost from the iTunes Store Purchased screen. The link to Purchased is on the iTunes Store Home screen, under QUICK LINKS. You need to be signed in with the Apple ID used to make the purchases.
There are third-party methods and utilities that can transfer song files from iPod to computer. If you do a Google search on something like "ipod transfer," you should get some links.