How can I know exactly which songs I have purchased from iTunes?
The trick is to navigate to iTunes new interface for albums (not search results) (e.g. by clicking the album's name). Instead of seeing the song price, you will see the word "Purchased".

To get to the song's album, you can either search for the artist and try the 1-3 likely albums the song might be located in until you find the "purchased" text, or you could click the "Write a Review" link for the song in your email receipt. Note that when you click the "Write a Review" link, you will be taken to iTunes old interface for albums. You must click the album name at the top to get to the new interface, which shows the "Purchased" text.

Another thing to note is that I noticed that if a song you purchased belongs to an album with only one song (i.e. it's the only song in the album), the "Purchased" text won't appear.