No, a single nucleus either emits alpha particle or beta particle at a time, but not together. If the nucleus is left in the excited state after the decay, it may emit gamma ray. Hence, a single nucleus cannot emit alpha particle, beta particle and gamma ray at a time.
Cosmic rays are high-energy radiation, mainly originating outside the Solar System. They are primarily composed of high-energy protons and atomic nuclei. They have mysterious origin.
The unstable nuclei decays following the law of conservation of energy and charge. If proton or neutron is emitted, these laws are violated or may require additional energy for emission. But emission of alpha and beta particle in the decay of radioactive materials are energetically possible. So, heavy nuclei usually decay by emitting an alpha or beta particle.