Felipa, you're right. Although I grew up singing this at every Seder and just assumed it was a Jewish song (as one does), I did some googling and found this NPR piece explaining how American Jews adopted/culturally appropriated the song: How The African-American Spiritual 'Go Down Moses' Became Part Of Passover

As for Chad Gadya, the Haggadah my family used said that the kid symbolised Israel/the Jews, victimised by Egypt and all the other countries/religions/cultures (I'm paraphrasing). Also, the penultimate verse refers to the Angel of Death, who plays an important role in the Passover story.

But I'd guess that the main reason it's part of the Passover service is that when you have small children who have to sit through a meal that can take several hours and is full of lots of solemn, religious material, a song like that is useful to regain their interest and allows them to participate and maybe even get a few wiggles out of their system.