The energy of the jump comes from the energy stored in the deformation of the popper. The gravitational potential energy merely serves as the activation energy required to unleash the stored energy. In fact, the height of the jump does not change with the height from which the popper was dropped.
Apparently, chemistry students are familiar with the concept of activation energy. As a physics tutor, this toy reminds me of the energy required to trigger nuclear fusion or fission.
Alpha particles are highly ionising, but have very little penetrative power as they lose their energy rather quickly. All it takes is a few cm of air or a piece of tissue paper to completely bring alpha particles to a stop.
Beta particles are less ionising, but more penetrative compared to alpha. Depending on how energetic the beta particles are, it a few mm of aluminium to soak up the beta particles.
Gamma radiation are really EM radiation. They are only weakly ionising, but have very high penetrative power. It takes a thick piece of lead to stop gamma.
With these facts, it is clear that this radiation consists of predominantly alpha particles. There rest are probably beta particles, and perhaps even gamma radiation.
The source is actually Americium-241 obtained from a smoke detector. Am-241 is an alpha-emitter with a half-life of 432.7 years. It does emit a little bit of gamma radiation. Its daughter nuclide Neptunium-237 is a alpha-emitter with a half-life of 160 000 years. The grand daughter nuclide Protactinum-233 is a beta-emitter with half-life of 27 days.