This indicates that soil in Arizona should be good:
Compressed Earth Blocks - Joseli LLC
It says to perform the jar test, which is described here:
EarthTek Inc
If soil expands, it is unsuitable.
That would be easy enough to test.
From what I have been able to tell, an underground shed is an over-built above-ground shed--below ground.
I naively thought that being below ground would provide insulation and thermal mass, but no, I am supposed to provide that myself, plus endless waterproofing.
This Old House made a video about how to make a shed. They did not say how long it would last. They provided a materials list, but they did not seem to enumerate everything visible in the video: