Quote - CTRL+E>loopcut or hot key K  for the knife tool to cut across existing edge loops.

 
Thanks for the tip. I think that was somewhere in the back of my mind, but I got the crazy idea of connecting the vertices with F. I didn't want to do that, though, because it would have been too tricky and time-consuming.

Quote - Start over and this time don't use a Curve circle.  Start with a Mesh circle (reduce to 16 verts) this will allow you to control how the faces and edges are made.

I'll try that, too.

It is better to do one thing well, than to do many things and excel at nothing.