I wouldn't have thought falling off at the rear was possible... is it falling off the smallest cog to between the cog and frame?
Is the derailer pulling properly to the back?
Looking at chain length is not a bad idea. Put it on the biggest sprocket and make the chain as short as possible.
Actually, the chain catcher will only work when it is in the highest gear. This is what I would do: 1) shift it to the highest (hardest to pedal) gear; 2) adjust the chain catcher so that it is riding about 1mm above the chain.
Now, when you fold, it should catch the chain and keep it on the bike. You have front chainguards, right?