Why are my songs not in my iTunes when I log in under my same ID and how can I get them?
Why would they be? You didn't back them up anywhere and iTunes does automatically redownload them to new library once you have authorized it. You can redownload most iTunes purchases again at no cost using the instructions in this article.
As for the error message on your iPod, it sounds like it's hard drive is dying. See this Apple support document for more information.
You may also want to take a look at the extensive post from Mitch 751 from more potential solutions/fixes.