When two oppositely charged particles are brought closer together under a Coulombic potential, the potential energy of the pair decreases. However, conservation of energy requires that this potential energy be converted into some other form (total energy is conserved). In a simple classical description we say that it is converted into kinetic energy, ie $-\Delta E_{Coulomb} = \Delta E_{kin}$, but the total energy remains constant, unless some of it is transferred to some other body. In the quantum mechanical description of atoms, electrons occupy discrete states, each state characterized by a total electronic energy (with potential and kinetic contributions) and a distance distribution between electron and nucleus (which can be used to compute an average distance). Transitions can occur between states through the exchange of discrete amounts of energy (quanta) with other bodies.
An ion formed by a neutral atom and an electron bound at a large separation is in a high energy state relative to the ground state, the relative energy being nearly the ionization energy of the ion. This is a highly unstable state because the slightest perturbation can drive electron and nucleus apart into an unbound (free) state. For the ion to settle into a lower energy (more stable) bound electronic state, corresponding to a smaller average distance between electron and nucleus, it has to release energy. It might do this radiatively (emitting photons) or through collisions with other atoms, dissipating the energy as heat. Another way of seeing this is that the newly formed ion has an excess of energy that it can give away to colder atoms in order to relax into a more stable lower energy state. QM dictates what are the most stable arrangements of electrons about nuclei, and only particular arrangements are possible. It is largely a geometric problem involving a balance between electron-electron repulsions, electron-nuclear attraction, the wave nature of matter evident particularly at small scales, and odd effects such as Pauli exclusion (the impossibility of two electrons having identical properties).
Considering now your list in more detail:
It doesn't always do so. Sometimes it's impossible. Not all combinations are stable. Only when the electron affinity is consistent with electron attraction is a stable combination possible. This is largely addressed above. Assuming it is possible to form a stable atom or ion, the combined particles have to shed energy to the surroundings, otherwise they risk falling apart again. When it gives off energy the combination relaxes from an excited state to a lower energy state. It might only shed some energy and remain in a highly excited (reactive) state, however.
This is addressed in another answer to your post. The argument is somewhat circular. Higher energy is less stable. Basically you are creating an unstable (excited) system when you attempt to remove an electron from a stable atom, so you have to add energy to do so.
This is addressed by the hand-wavy QM argument at the end of my dissertation. The combination of charge attraction and repulsion with the wave nature of subatomic particles and the Pauli exclusion principle lead to particular particle combinations and geometric arrangements being stable. This is the subject of a more rigorous QM course.