The one song is "Elle Me Dit" by Mika, and Babette's ring tone is the popular children's song "Alouette", author unknown.
Regarding the first song: Originally, I wanted to take another song. But unfortunately, the only things I could remember about it were the melody, and the fact that it was in French. "Elle me dit" is at least conveying a mood halfway similar to that of this mystery song.
I thought some of those lyrics looked familiar, I've only ever heard the English version of the song! I had no idea I was listening to a translation :o
I remember that I spent quite some time with choosing a song that would fit Babette. Because, like I said in the author comment, I had no idea how to find the song I originally had in mind!!
* Oh boy, from here, there're still a lot of chapters before she gets introduced...