Why on iPad or iPhone, the $0.99 click is so easy, but to download my purchased songs, it is so XYZ tough?
I believe if the song is present on your iPad/iPhone library it will be grayed out in iTunes (on iOS) probably the purchased playlist?

In regards to that downloading again without repurchasing, ONLY applications purchased can be re-downloaded for free without repurchasing. Songs and audiobooks require you to re-purchase. This is because the record companies have explicitly stated this in their agreement to Apple. Whoever told you that information was wrong.

However, if you email Apple, you can request to restore all downloads. This means everything you bought will be available for download again (except some that have changed name or have been removed from iTunes store) but do note that this can be used one time only. It's for people who had their laptop stolen and lost everything or a broken hard drive with loss of data.

Hope that clears things up.