As the title says, what was the first written piece of music we know of in what we call the new world? What was it written for? Are there recordings?
My answer focuses on music written in the "western" and "art music" traditions. If you're looking for folk music or art music composed by indigenous peoples, that would be super awesome - I'm not personally aware of any, but then it's not really my area of study either.
The answer you're looking for may well be in the Oxford History of Music series by Richard Taruskin. It's...surprisingly readable for a textbook about the history of music, though I haven't actually reached the chapters that talk about new world music yet.
There are plenty of recordings of "Latin American Baroque" music out there, which is exactly what it sounds like. The Baroque period runs from about 1650-1750. Almost certainly most of the extant pieces from this time period are going to be church/sacred music.
I can't exactly answer your questions but...
The oldest known example of printed music in the new world dates from 1556. Juan Pablos published an Ordinarium in Mexico. It contained some chants (but no polyphony) that were meant to be the correct version of what should be sung for religious ceremonies.
Hernando Franco (1532-1585) was one of the first composers I now that were active in what is now Guatemala and Mexico, so his music should give you a decent idea of what was going on.
That sounds like Spanish polyphony because, well, he was born and trained in Spain...
If you want to listen to something a little bit less Spanish, here's some music by Juan Gutiérrez de Padilla (1590ish – 1664). That is a "negrilla," a villancico that was trying to portrait some elements of the music that the African slaves brought to the new world. That kind of music used some dialect of Spanish and some words from the languages spoken by the slaves. This music apparently used instruments that were not brought by the Spanish conquerors. As with any other form of very old music, we are not really sure how it exactly sounded and recordings of modern performances can't really be taken as "this is what it sounded like."
Search for "música virreinal" and you will find LOTS of recordings. In the last 30 years this music has become much more studied and performed.
Also, there was music way before the Spanish conquerors arrived. However, I am afraid most of it is lost forever. Some people have tried to sell some stuff as prehispanic music, but as far as I know it's all made up bullshit.