I want to take a photo and use the stored GPS coordinates and the orientation of the phone to do some calculations. How can I access that information?
Alternatively if there is an API to do that I could make my own app.
Of course! You can use Metadata viewer to see the GPS coordinates and other information of your photos. Android has Handful of APIs for anything you want. If you a developer you can make use of that. Here is all you need.
This is one of those problems that I believe requires a documented standard: the same reason every platform trivially accesses the GPS location of a photo is because of EXIF and it's standardized tags.
EXIF is not user-extensible, and side car files are too namespace-y (proprietary).
Unless I'm wrong about there not being a standard place for Euler angles or even the quaternion that establishes device attitude.