False, you can write your own API without permission from AMD, they freely allow that. APIs are just abstraction layers to make software development easier for cross platform and multi device development.
Yes API are easy to write for a software programmer.
No, PS4 has 2 APIs;
GNM - Low level API that allows you to write directly to GPU meaning you can create a mesh shader implementation yourself for your own specific game engine
GNMX - Is very similar to DirectX and Vulkan. Anything DX can do, GNMX can do. PSSL is Sony's shader language similar to Microsoft's HLSL
Yes APIs are important for platform and hardware agnostic development. It makes things non-vendor specific. You know who is a part of Khronos and a Promoter member? Sony.
You are clueless, you know what opengl and vulkan has in common? They are both made by Khronos Group.
OpenGL supports ray tracing, but it is a vendor specific extension made by IMGTec for their hardware.