I might be able to answer this.
So I believe the requirement is that apps must run on standard (meaning mid-range) smartphones and it will be up to the tested demographic to provide the handsets - thus we don't mandate which phones they are being tested on. All smartphones have a microphone (kinda important for calls ) but for headphones it will be dependent on whether the testing demo has headphones to use.
In terms of code sharing, there is no Open Source requirement for the Adult Literacy XPRIZE and teams can use whatever platform they like. I believe any code sharing is fine - teams are welcome to utilize Open Source in their projects.
(this is all based on my understanding... @edmcnierney may correct me)