I've solved it once so far. But I'm not sure if I got lucky or not.

Here is my general solution plan for the regular Heli Cube. When using these steps, is it possible to end up with 2 or 4 edges on the top face flipped?

1) Solve the edges and centers on the bottom face.
2) Solve the corners on the bottom face. (For the Curvy Copter, while doing this, make sure the edges between the bottom and top halves are solved at the same time.)
3) Solve the rest of the bottom half. (When finished, the four middle edges should still be solved too.)
4) Solve the corners.
5) Use one-orbit-3-cycles and two-orbit-2-cycles to finish solving the centers.

My 3-cycle algorithm uses each edge in it an even number of times, so it doesn't change the edges on the Curvy Copter. But my double-2 cycle uses each edge in it an odd number of times, so it does change the edges. So, what I want to know is, if I have to use the double-2 cycle, will there always be 2 edges flipped along with it? (For my first solve, there were.)

(This was without jumbling. I haven't done a full jumble scramble on either it or the regular Heli cube yet...)

If all of the corners have been placed into their correct position then the number of flipped edge centers is always even.

It's pretty easy to see why. All corners positioned correctly is an even permutation. There is no way to change the orientation of a single middle edge without also changing the permutation of the corners. This means the overall orientation of the middle edges is tied to the overall permutation of the corners.

This is even true with jumbling although I have a suspicion that the additional jumbling moves on the CC+ will break this linking.

It's guaranteed to be even, but not necessarily related to the positions of the "petals"?

I personally solve the bottom half first (including equatorial edges), then solve rest of the corners and orient edges and use petal 3-cycles to finish it.

And two 3-cycles can make a double 2-cycle (except without inverting edges), so edge orientations aren't related to petal positions, if I understood right.