[Work in progress. I may decrease the number of diode branches for simplicity.]
Progress has been made on the sine wave output. Apparently you can use a technique called diode wave shaping to turn a triangle wave into a sine wave. Here's the paper I found: http://tinyurl.com/lew34ez
Here's the schematic I went with:
Here's a useful graph:
And here's another:
In the first graph, each of the red x are the points in the wave when another diode stage is turned on. The slope of each line segment is directly related to the voltage division ratio.
In the second graph, I called the curve that my circuit is being designed to follow the shaping curve. I basically chopped a quarter cycle sine wave into four equal segments and linearized each segment.