supposedly you can play or download songs using the icloud download button, a cloud icon with a downward arrow inside. But nothing happens when I click or double click on it.

You have to have music in your iCloud account to do that...when you open the Music app on your iPad, for example, you can access all music from iCloud that you have already placed in the iCloud or associated with iCloud. That is not for downloading music you have not already purchased or downloaded.