Can't play music on iTunes

Repeated requests to authorize an already authorized computer for store downloads typically indicates you have corrupt or incomplete copies. Backup and then delete the tracks that won't play from your library without hiding from your purchase history. Redownload using the cloud links that should remain, or close iTunes and reopen if they don't show up straight away. While downloading click the activity widget and turn off the option that allows for simultaneous downloads.

If that hasn't helped see If iTunes for Windows prompts you to authorize your computer when you try to play purchases - Apple Support.

Regarding the old ID see If you forgot your Apple ID password - Apple Support. You can authorize your computer for multiple IDs.

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Prior to 2010 Apple sold DRM laden tracks on a one time only download basis. You didn't get to redownload them, and Apple weren't necessarily keeping a record. The ability to redownload past purchases is also dependent on the content still being in the store. Sometimes rights holders remove content.

It might vary somewhat by version of iTunes but when you delete a purchased item you should get a prompt asking if you also want to send the deleted item to the recycle bin, and then possibly another prompt asking if you want to hide the purchase from your purchase history. You would say no to this second prompt. Current versions of iTunes will show cloud links for any unhidden purchases that you haven't downloaded, which will let you stream the item from Apple's servers, or download it to your library. You can use these steps to refresh you copies of purchases that won't play correctly, but that is only going to help if the content is visible in the purchase history. For older items you might have to treat those the same way you would a lost physical disc, replacing with a new purchase if you want to.

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