No one ever said anything about "following a definite path." The photons
emitted by the light in the double slit experiment SPREAD OUT.
But each of your individuals "photons" does not spread out. You admit this when you say "Each travels it OWN path away from the emitter". You see? Each of your "photons" follows some specific path. When you say "the photons spread out" you mean some photons follow slightly different paths from the others, you don't mean that any individual "photon" spreads out... and this is why your beliefs cannot explain the observed interference pattern. To explain the interference pattern with individual photons you need some kind of "spreading out" of individual photons, because the behavior of the individual photons that go through (say) the Left slit depends on whether or not the Right slit is open. Quantum electrodynamics explains this by the "sum over paths" (as described in detail in the previous message), whereas your concept of a "photon" cannot explain it at all.
Now, close one of the slits. We find that the striped pattern disappears, and photons can land at all the locations, including the spots where no photons ever land when both slits are open. This means that when a photon passes through one of the slits, it is affected by whether the other slit is open.
No, it is affected by the fact that it has only one route to the screen.
Then you cannot explain the interference pattern when both slits are open. This was the whole point of the experiment, going all the way back to 1800, and this is why your concept of a "photon", a la Newton's "oscillating" particles ("fits and starts") had to be abandoned, because it couldn't explain interference effects. The particle theory of light was only resurrected when quantum effects were discovered, leading to the "sum over paths" approach. That's why a photon is not a classical particle, whereas your conception of a "photon" is a classical particle, because each of your "photons" has an individual singular path.
If a photon does not oscillate, what is the difference between red light
and blue light?
Each photon of blue light has more energy than those of red light, and the phases of the sequence of photons (along a single path) comprising blue light advance more rapidly than for red light. The phase of an individual photon for an individual path doesn't change in flight, but a sequence of photons have different phases, corresponding to the phase of the source at emission. Also, when multiple paths are available for a single photon, with different path lengths, the probability is given by the magnitude of the sum of the phased amplitudes, which is why the interference effects occur for individual photons, dependent on the frequency of the light.
When photons can go through two slits, they can divide into two patterns. When photons can
only go through one slit, they do not divide and form just one pattern.
That's the classical image, which can't account for the interference pattern. You see, with your concept of a "photon", each going through just one slit or the other, you would get a smooth (not striped) pattern with just the Left slit open, and you would get the same smooth pattern but shifted slightly to the right if just the Right slit is open, and if both slits are open you would get the sum of those two smooth patterns. You would not get any interference effects. That's why Young's experiment killed the Newton/Lake model of the "photon" 200 years ago.
Remember, when both slits are open, a sequence of individual photons (possibly hours apart) reach the screen, and each one lands in a specific place (not smeared out), but if we keep track of where each one lands, and make a histogram plot, we find the "striped" interference pattern of dark and light stripes, no photons ever land in the dark stripes. If we close one of the slits, we find that the striped pattern disappears, and photons *can* land at all the locations, including the spots where no photons ever land when both slits are open. This means that when a photon passes through one of the slits, it is affected by whether the other slit is open. That is why the Newton/Lake model of the photon is invalidated by experiment.
What appears to happen with the Double Slit experiment is that the
photons get POLARIZED.
Polarization (with or without caps) does not account for the interference pattern that emerges for a sequence of individual photons (or electrons) with both slits open, and the disappearance of that pattern when only one is open (or when we detect which slit an electron passed through). Also, please note that the same experiment with electrons (massive particles) gives the same interference results, so the answer is not unique to photons.
Any viable theory must be able to explain the following basic fact: There are locations on the screen that can be hit when just one slit is open and the other is closed, but that cannot be hit when both slits are open.
This applies not just to photons, but also to any massive particles (e.g., electrons) as well, since all particles exhibit wave-like behavior with interference.
Working on the paper seems a much more productive way to find
answers to questions...
But you've already been provided with the answers to all your questions - and more. If there was something in those answers that you think is wrong or unclear, you could ask follow-up questions. That's the most productive approach. You're not going to re-discover quantum electrodynamics on your own - apparently not even after having read fairly detailed descriptions of it.
How can something be considered a "mystery" for hundreds of years
when there doesn't seem to be any mystery about it at all?
It can't. Also, the mystery is not hundreds of years, it's less than 100. When Young first performed his experiment it wasn't mysterious, it simply led to the recognition of the wavelike behavior of light, which is perfectly classical, and hence refuted the Newton/Lake classical particle idea for light. The modern mystery arose only in the 20th century when the photo-electric effect and other phenomena eventually led to the modern concept of the photon (completely unlike your toy concept), which requires the "sum-over-paths" approach (or something equivalent), which is non-classical.