1) FAQ says: "The sun and moon, each 32 miles in diameter, rotate at a height of 3000 miles above sea level."
* If the sun was 32 miles in diameter, it would not have enough gravity to sustain nuclear fusion (AFAIK the FE model does not dispute General Relativity).
* We can use trigonometry on a flat surface to calculate the distance to the Sun and verify that it is a lot more than 3000 miles. Perform the following experiment: Take two points 5.2 miles part. Call the points A and B. From point A, measure the angle between B and the sun. From point B, measure the angle from A to the sun. If the Sun is 3000 miles away, basic trigonometry tells us that there should be a difference of 0.1 degrees in the two measurements. This discrepancy is easily detectable with standard nautical or topographical equipment.
2) FAQ says: "Each functions similar to a 'spotlight,'"
Light sources do not behave like a spotlight. A spotlight behaves the way it does because humans put the light source inside a container that purposely blocks light going in other directions and usually also reflects it in the desired direction. To suppose that the sun and moon behave like spotlights, requires the assumption of similar containers designed to block light going in other directions.
The spotlight hypothesis is demonstrably false. When you get further away from a spotlight that is not pointing directly at you, you don't see a round spotlight sink on the horizon. What you see is the circular light become oval shaped, with the oval getting thinner until no light is visible. This is not what we observe in the real world. Hence, the sun and moon are not spotlights.
The spotlight hypothesis is also contradicted by eclipses. If the moon was a spotlight, during a solar eclipse we would see the moon as usual, rather than a black circle. The reason we see a black circle is that the moon produces no light of its own.
The spotlight hypothesis is also contradicted by lunar eclipses. Why does the moon change colour and then go dark at precisely the points that RE theory says that the moon enters the penumbra and umbra?
If the sun-spotlight is hovering around the equator as in the FE model, that would mean that days are longer in the equator than in other latitudes, and that days are longer in the northern hemisphere than in the southern hemisphere. We know both ideas to be false. Furthermore, the southern hemisphere would get less over-all sunlight, making it much colder than the northern hemisphere.
If the sun hovers above the equator in the way described by the FE model, then an observer on the equator would see the sun make an arc across the sky. This is not what we observe. A person on the equator sees the sun go straight up, to the zenith and then straight down. In higher latitudes you see the sun roll across the horizon a bit more (I have lived both next to the equator, and at high latitudes). This observation contradicts the FE model.
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Edited to add: Another problem with the FE model is that it does not explain through which mechanism the moon is supposed to be a light source for "cold light". Furthermore, if you grab an amateur telescope and point it at the moon, you will see surface features like craters. These are visible because they cast a shadow. How can they cast a shadow if the moon itself is the light source? Finally, any amateur astronomer knows that the best place to aim your telescope is near the edge between the light and dark sides of the moon, because that is where the shadows are biggest (making surface features more visible). How is this possible if the moon itself is the light source?
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3) FAQ says: "The stars are at a height of 3100 miles above sea level, which is as far as from San Francisco to Boston."
The same parallax experiment explained in (1) also disproves this.
The fact that different stars are visible at different latitudes conflicts with the FE model. In a round earth, the earth obscures parts of the celestial sphere from different observers (you can't see Polaris from Australia, and you can't see the Southern Cross from Sweden). On a flat earth, the planet does not obscure different parts of the sky for different observers. Everybody gets the same sky.
The fact that during the course of the night the stars rotate around a point in the sky (e.g. Polaris) and this point moves up as one moves to greater latitudes also conflicts with the FE model.
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Edited to add: The FE model also fails to account for the fact that different stars are visible at different times in the year.
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4) FAQ says (on the topic of sunsets): "It's a perspective effect. Really, the sun is just getting farther away; it looks like it's disappearing because everything gets smaller, and eventually disappears as it gets farther away."
The sun does not get smaller during sunsets. The angular size of the sun can be easily measured (compare it to a dime held at arms length).
5) FAQ says: "Dark Energy accelerates the Earth and all celestial bodies in the universe at 9.8m/s^2"
This dark energy requires justification and supporting evidence. It is entirely unlike the dark energy that astrophysicists know about (I have a degree in astrophysics). I suggest using a different name for this mysterious energy, lets it be confused with the dark energy used in astrophysics literature.
6) FAQ says: "FE assumes that the Earth does not generate a gravitational field."
Theories that predict that the laws of physics are different for Earth have a tendency to be wrong. They also defy Occam's razor and the mediocrity principle. Second, variations in Earth's gravitational field have been measured (e.g. gravity is a tad weaker in Canada) and are consistent with Earth being a round planet that generates a gravitational field just like all other objects. The gravitational field can also be measured through atomic clocks, since gravity makes time slow down, according to General Relativity. These experiments indicate that clocks run slower at sea level than on an air plane. This is consistent with Earth generating a gravitational field, and inconsistent with the lower perceived gravity being a result of celestial bodies exerting a gravitational influence.
7) FAQ says: [gravity varies with altitude because] "The celestial bodies have a slight gravitational pull."
This is contradicted by experiments with atomic clocks. Time runs slower closer to the Earth. If the lower gravity was due to the celestial bodies exerting a gravitational pull, the clocks would not behave this way.
FAQ says: "There is a field created during the interaction between Dark Energy and the Earth. This is known as the Dark Energy Field, and it acts as a containment to prevent DE from affecting the objects on Earth. This explains why the atmosphere will not be diffused into space"
This also requires explanation and supporting evidence. This must be a very magical sort of "energy" that makes air stay inside the FE. Calling it a "field" doesn't explain anything.
9) FAQ says: "The gravitational pull of the celestial bodies provide tidal effects."
This fails to explain why there are two tides per day and why they follow the moon and not the sun. The "sun-moon" thing is yet another unproven postulate to try to force the FE model to fit the evidence. When you keep adding unseen, unproven, and unprovable hypothesis to a model, that is a sign that the model is flawed.
Even if we allow the "sun-moon" object proposed in the FAQ, the FE model still fails to explain why the tides don't point to the moon, but seem to come a little bit behind it. In the RE model, Earth's rotation explains the gap between the moon's position and the tides.
10) FAQ says: "The sun circles over the equator, thus the poles don't receive the same intensity of light."
If the sun is 3000 miles away and circles the equator, the incident light on the poles would be at 65 degrees. That is not nearly low enough to explain the temperatures at the poles. For comparison, the city of Orlando, Florida, gets incident light at 62 degrees. If you live in Canada or Europe, it is easy to see that the average sun position is at less than 45 degrees from the horizon. If you live in Sweden or Norway, the sun is even lower. At the north pole, the average incidence angle is zero. This contradicts the FE model.
11) FAQ says: "An undetectable celestial body, known as the antimoon, passes between the sun and moon. This projects a shadow upon the moon [causing lunar eclipses and moon phases]."
Another undetectable object. How convenient... Anyways, this cannot work. During moon phases, part of the moon is entirely black. If it was a shadow, the moon would be inside the umbra, making the moon darker but not black, as in a lunar eclipse. Second, this explanation is inconsistent with the earlier claim that the moon is a spotlight. If the moon produces its own light, then the idea of casting a shadow on it makes no sense. Unless of course the moon is not a spotlight and its light is reflected, in which case you need to explain how it can reflect sun light if the sun is a spotlight and you need to explain why moon phases make some of the moon black, but lunar eclipses make it red.
12) FAQ says: "The airline pilots are guided by their GPS."
Air planes and airlines are much older than GPS. Even today, not all air planes use GPS. And for hundreds of years, ships have sailed the oceans using only stars and a round earth coordinate system to navigate around the oceans, including many trips around the southern hemisphere from Europe to Africa to India, to China and Australia.
13) FAQ says: "The magnetic field is generated in the same fashion as with the RE (Diagram)."
This is flawed. Earth's magnetic pole does not coincide with the geographic north, and indeed, the magnetic poles are in constant movement. This is fine in a spherical earth, but more problematic in a cylindrical earth.
14) FAQ says: [flushing toilet] "On a round Earth, the Coriolis effect adds at most one (counter)clockwise rotation per day."
This is a straw man argument. The Coriolis effect has many significant terrestrial effects a lot more important than flushing my toilet. The Coriolis effect has significant meteorological effects such as cyclones, air currents and ocean currents. Air tends to move toward low pressure regions, but not on a straight line. On a non-rotating frame of reference, air should move in the direction of the pressure gradient. Instead, large scale air movements are perpendicular to the pressure gradient. This is known as geostrophic flow. There are many other important terrestrial effects caused by the Coriolis effect. Please look up "Coriolis effect" in Wikipedia.
Earth's rotation (and the Coriolis effect) is also easy to demonstrate using a pendulum (look up Foucault's pendulum). I have done this experiment myself.
Earth's rotation is also easily observable on a clear night away from the tropics, when you can actually observe the stars rotating around a fixed point in the sky. You can see that the position of this point corresponds to your latitude, so that, for example, at the north and south poles, Polaris is directly above you, all the time, and the stars all rotate around it.
15) FAQ says: [seasons] "The radius of the sun's orbit around the Earth's axis symmetry varies throughout the year, being smallest when summer is in the northern annulus and largest when it is summer in the southern annulus."
Another unexplained phenomenon. And once again, one that doesn't solve anything. This hypothesis can begin to explain why the sun's angle changes during summer and winter, but it fails to explain why days are longer in summer and shorter in winter. This is also a good time to again point out that the spotlight model of the sun, as in the FE model, would mean that over the course of the year days are longer in the equator than at other latitudes, and that days are longer in the northern hemisphere than the southern hemisphere. So the spotlight hypothesis contradicts observations.
16) FAQ says: "NASA and the rest of the world's space agencies who claim to have been to space are involved in a Conspiracy to keep the shape of the Earth hidden. The pictures are faked using simple imaging software."
I have personally met two people who have flown into space. One was an astronaut, the other a space tourist. Maybe the astronaut was lying to me when she showed me her pictures at MIR, but the space tourist is someone who has no reason to lie to me about his going to space and seeing the earth being round.