Wow! Nice solution. I have been thinking about it for a while and I still have two open quiestions:
The first one seems to be true (at least the cases tested) but I can't see why. I am not so sure about the second one.
Can you give me any clue about them?
Due to my background I would go for a graph search based solution. The nodes of the graph are the state of the row of pancakes and the neighbors of a node are those states reachable by a flip action.
It is not the fastest one, but it is exhaustive and the solution is optimal.
Does it work in any case?
I cannot show you the mathematical explanation but I have some hints.
Let's take the example of the post but reversed:
These are exactly the same flips made above.
I think the key point here is that you should never touch happy pancakes, unless forced to do so because of a nearest unhappy one.