What's the name of rays and faces in high dimensional spaces?

"Half-space" is the generalization of "ray" (aka, "half-line").

For corner-like analogues of angles, I'm not sure of a particular name for arbitrary cases. However, "orthant" is the generalization of the $2d$ "quadrant" and $3d$ "octant": a region bounded by $d$ mutually-perpendicular hyperplanes through a vertex (usually, the coordinate hyperplanes through the origin). Note that we can assign a "solid angle" measurement ---in "steradians"--- to corner-like regions, even if the boundary elements aren't "flat".

For regions fully bounded by hyperplanes, the generalization of "point", "segment", "polygon", "polyhedron" is "polytope". A $d$-polytope is bounded by $(d-1)$-polytopes: endpoints of a segment; edges of a polygon; faces of a polyhedron. The generalization there is often "facets of a polytope", although "facet" is also used to describe certain non-face elements of a polyhedron, so be careful.