Here are few ideas:

1. I like the idea of a course about polytopes. Few books but some are excellent: "Lecture on polytopes" by Ziegler or "Convex polytopes" by Grunbaum are the obvious choices.

2. A course about curves and surfaces + an introduction to manifolds should satisfy 1-6 without troubles. "Differential geometry of curves and surfaces" by Do Carmo is a very good book; there are plenty of excellent books about manifolds.

3. A basic course on algebraic varieties require the use of algebra and differential calculus and gives example of spaces with pathological spaces (i.e. non Hausdorff and/or with singularities)

4. I guessed you want a more modern geometry course but without leaving the view of the formation of high school teachers. Michèle Audin wrote a very good book about affine, projective, curves and surfaces. It is aimed to future (French) high school teachers. I guess the title is "Geometry" (it is "Géométrie" in the French version).

I don't know the curriculum of a typical American student so I hope my suggestions are still pertinent (especially the point 3).