"Simplex" is the $n$-dimensional generalization of the $1$-dimensional segment, the $2$-dimensional triangle, and the $3$-dimensional tetrahedron.
"Hyperplane" is the generalization of the $0$-dimensional dot in $1$-dimensional space, the $1$-dimensional line in $2$-dimensional space, and the $2$-dimensional plane in $3$-dimensional space.
Are there names for generalized rays and faces, i.e., parts of hyperplanes with one or more boundaries in $n$-dimensional space?
"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.
One can generalize ray to half-plane and, in higher dimensions, to half-space.
Faces can be awkward. Different mathematical disciplines can use words such as polyhedron, polytope, face, facet and so on in different and sometimes incompatible ways. Typically the higher-dimensional generalization of a face will be either a face or a facet - terms such as cell or hypercell are also occasionally still used.
