All the confusion comes from mixing terms. The phantom has a service ceiling of 6000 m. That's how high the bird can physically (or mechanically in the terms the OP used) fly. It doesn't matter if that height is over the ocean or in the mountains of Nepal. Firmware limits the height to 500 m above the point of take off, again regardless of the altitude of the takeoff point. Legally, proper authorities can legally limit the maximum height (referenced to the ground below the aircraft) such as the 400 agl limit for Part 107 pilots. All of the above are correct, you have to decide which question you are asking.
I'm not so sure about that. I think you are quoting statistical limitations without considering the physics.
I find myself wondering if the hardware itself is capable of flying 6000M above the peak of Everest...
At some point there won't be enough air for the propellers to displace in order to compensate for the weight of the craft, and the battery function at that temperature may either be so dramatically reduced as to be non-functional, or at the very least incapable of sustaining a 6000M climb.