You make many questions in one, all of them have their own answer.
* Just to clarify, nuclear decay and nuclear reaction are two totally separated and different things. Radioactivity occurs naturally, spontaneously. You have to sit and wait for the nucleus to decay. A nuclear reaction is forced, is something you obtain by, for example, shooting a particle beam into a target. I am going to explain radioactivity, because nuclear reactions is something wide and huge.
Yes, everything gets "quantum" when you go to very small contexts (microscopic contexts, like atoms, nuclei, particles, etc).
Nucleons inside the nucleus (i.e., protons and neutrons) are bounded particles (bounded by the nuclear potential) organized in quantum shells, in a similar (but not equal) way as the electrons do organize inside an atom. Nucleons are well described by quantum wavefunctions and their associated quantum numbers (energy, spin, nucleus and particle angular moments, etc.). The energies separating those shells are quantized, which means, you can only move through shells (or energy levels) by absorbing or radiating energy (or particles) in discrete values. In some contexts, you can see a particle as a form of discrete energy, the most common example is the photon (particle) = gamma ray (energy). 969
In nuclear decays, there is always some gamma emission happening (unless you decay directly to the lower state of the daughter nucleus). These energies are generally solutions of the Hamiltonian of the system, if you are able to solve the Hamiltonian in the first place, which is not an easy problem, especially for exotic and unknown nuclei.
These energy values separating shells are not the same for the same shells in different nuclei, because when you start adding and removing nucleons, the interactions between them change, and that affects the nuclear potential, which in turn affects the energy separation of the levels (the position where the shells lie, to say it in a "human" language, which get pushed upwards or downwards). When the energy separation between shells increases, that may be a sign of higher stability (because you need more energy to move from one shell to the next), while with short energy separations it's so easy to jump to the next energy level.
Radioactivity is caused because the nucleus gets unstable. There are many reasons why the nucleus can get unstable, and it will try to get rid of that instability and reach the lowest energy state of the system (in this case the system is the nucleus), like everything in Nature.
The common scenario is: you have an unstable parent nucleus that decays. Depending on the case, it emits some particles.
Parent Nucleus -> Daughter Nucleus + particles or tiny nuclei + energy
Some of the energy released in the decay is used as kinetic energy by the rests of the decay, but some energy may be saved as excitation energy in the daughter nucleus. Since the nucleons in the daughter nucleus are organized in a certain way according to shells, quantum numbers, energies, etc., that nucleus will have a set of energy levels that is like a footprint, and the nucleons of that nucleus will be promoted to higher energy levels using the excitation energy we mentioned above.
An image is worth a thousand words, so please check this diagram of a well known radioactive reference source, 60Co, where you can see how the parent (60Co) decays into Ni, and Ni has a set of excitation levels where you can end up when coming from 60Co (its complete level scheme is much more complex, but if you are coming from a decay, you can only access or "populate" certain energy levels, while others are forbidden by quantum physics laws).
Source: http://www.nndc.bnl.gov/chart/getdecayscheme.jsp?nucleus=60NI&dsid=60co%20bM%20decay%20%2810.467%20m%29&unc=nds
When the daughter nucleus finds itself in an excited state, (one is at 1332 and the other is at 2158) it will try to get rid of the instability by emitting particles, in this case, gamma rays of precise (quantized) energies (826, 2158 and 1332). This is the footprint of Ni for THAT decay.
Instabilities in the nucleus can be produced by the imbalance between the number of protons and the number of neutrons. Because the protons repulse each other since they have the same charge, when you have huge nuclei with a huge amount of protons, you need to increase the amount of neutrons to add more nuclear potential to the system, and keep it bounded. That's why light nuclei have the same amount of neutrons as of protons, but heavy nuclei have some protons and a bunch of neutrons. 969