Object disappearing when swapping from one motion path to another

This one is confusing. I've tested in your file, and can see what you mean re triggers breaking. I tried changing the oval to a group (so it's three items moving along a motion path as a group rather than using states) but then the whole screen was blank when I tested - not even the rectangle at the base of the screen. So weird. Grouping the items in the jump and fall states seemed to help in your 'not working' slide - but both 'wings' appeared behind the oval in test regardless of them being placed in front in the state. All very strange. 

I have found a potential work around for you. Instead of using an oval and two triangles, create both versions of the shape in ppt and save them as images. When it's only one object changing and not adding in additional objects in the states it seems to work fine. So instead of the shapes, you essentially have an image with two states - one with wings up and one with wings down. You can then set the states to change when your 'IsFalling' variable changes to true or false. That seemed to work for me.