I have a student who is prepping for an advanced algebra class, which is one step beyond Algebra II. Conic sections are not covered. The parabola and the Hyperbola are, but more from an algebraic standpoint. Seems to me conic sections would be a good idea, since it unifies the various topics in analytic geometry.
Generally, seems to me that, in terms of the "how" vs "why" questions brought up in the recent discussion on polynomial division, students are short-changed on the why. Math teaching has become incredibly complex, and there is often little in the way of explanation. Reference to old texts from the 19th Century often reveals some interesting and more informative approaches that have been cast by the wayside. One such approach is the solving of exponential equations using continued fractions. I've had a lot of fun practicing that algorithm, which actually shows you one way of calculating logarithms.