Where is the origin that maps the base hexagons in Uber's H3 Hexagonal hierarchical geospatial indexing system to the globe?

where is the origin?

There is no single origin. The grid is based on a projection of a regular icosahedron onto the surface of the globe (using a spherical model).

Where is the part where it maps the stuff to the actual Earth globe? Is that hardcoded somehow in the codebase somewhere?

The orientation of the icosahedron is fixed relative to the Earth - you can see the center coordinates of each face here (these are, as you suggested, hardcoded in the codebase).

How do I know that when I use the library today, the 122 base hexagons I get from the API will have the same hash as the last time I used it?

All H3 indexes are stable - this is a function of the algorithm we use to create the index, and of the API guarantee we offer that we won't make any changes to the library that change cell indexes or geographic locations.

You can see more info on the methodology of the indexing function here.