A photon is the smallest possible amplitude increment of an electromagnetic field. It’s like one notch on the volume knob of your stereo.

So a photon can’t be built up, gradually increase in energy, because a photon is the smallest possible energy increment.

Your questions about where the photon starts and ends up traveling suggest you are imagining a trajectory for the photon. There is no such animal. One way to think about it is that when the atom emits a single, pure photon, the probability amplitude of electric field intensity over a certain volume of space suddenly jumps up one notch. When the photon is observed, the probability amplitude suddenly concentrates at whatever place the photon was observed.

What happened in between? That is not knowable, since no observations were made, and knowledge without observation is logically contradictory.

The mechanics of which you speak, the exact details of photon emission from an atom, would involve very complicated relativistic calculations, because an infinite number of complicated processes involving the electron(s), the final (real) photon, and an infinite number of virtual photons must all be considered. Even calculating in complete detail what happens when a single isolated electron is coupled to the radiation field (i.e. allowed to interact with virtual photons) is a problem for which Julian Schwinger and Richard Feynmann jointly received a Nobel Prize. And the problem was not entirely solved.

A photon does not expend energy to travel. This reflects Aristotelian physics, not Newtonian. You need to consider the meaning of inertia.

Entropy is not like energy. It is not a physical measurable quantity. Entropy is a measure of the ignorance implied by your definition of the state of a system. Hence depending on how you define the state of the system you are considering, you may have different values of the entropy. Generally most physicists interested in a simple treatment of photon emission would work with a single wavefunction, for which the entropy would be and remain zero.