Re: EPE of a neutral object?

Any charged object possesses a potential. When it is discharged, (just as you said) the flow of charges from the object constitutes an electric current (in fact, it is generally true that any potential difference causes a current to flow). As a result, the potential of the object falls. The energy comes from the fall in (electrical) potential energy of the charges. This is analogous to the situation in mechanics in which a mass falls from a height. During the fall, energy is released from the decrease in (gravitational) potential energy of the mass.

To determine the electrical potential of an object, you need to consider the charges held by the object, and also other charges present near the object, as these nearby charges would raise (or lower) the potential of the object in question.

A hydrogen atom as a whole is neutral, as an atom carries no NET charge. But inside the atom, the electron does possess a certain amount of (electrical) potential energy, as there is another charge (the proton) nearby. As you said, the energy of the electron is k(-e).(e)/r, or -ke^2/r (where k is the electrostatic constant, e is the electronic charge, and r is the orbital distance of the electron).

A hydrogen atom is stable. If left untouched, it will not decompose into a single proton and electron by itself. In other words, the hydrogen atom cannot "discharge" by itself. But when energy (of magnitude ke^2/r) is delivered to ionize the atom (i.e. removing the electron), the atom becomes "charged". It can be viewed as possessing electrical potential energy, as the "charged atom" can attract nearby electrons, the flow of which constitutes a "current" and energy is released. The electrical potential energy of the atom then returns to zero.

An analogous situation in mechanics is that the "neutral atom" is analogue to a mass placed on the ground. It is stable and possesses no potential energy. In removing the electron from the atom (ionization) to become an ion, it is analogue to raise the mass from ground level to a certain height. The mass then possesses some potential energy (analogous to the ion possesses some electrical potential energy).

When the mass falls back to ground level, energy is released and the mass becomes stable again, possessing no more energy. This is analogous to the ion neutralized by attracting a nearby electron (discharge process), releasing energy and becomes neutral again.

[Note: your saying that the system (i.e. the atom) possesses electrical potential energy of kq1q2/r is not wrong. In fact, it is the energy in FORMING the system (i.e. in binding an electron to a proton). But in general practice, especially in nuclear physics, such situation of a neutral atom is termed as the GROUND STATE, and my explanation given in the above paragraphs apply.]