"...looming is taken into account with the setting sun. It is what causes the sun to appear to flatten out as it sets..."
The flattening effect with the setting sun is caused by perspective, foreshortening occurs as the sun approaches the angular resolution limit of the observer. This is not a refractive effect."...you are using looming/refraction here to explain why the sun does not shrink as it gets farther away, but I think that your explanation raises more questions than it answers.
Can the effect be enough to account for the 3x magnification that would be necessary for the sun to be the same size when it is overhead and at dawn and dusk at the equator, and the 6x magnification necessary at the southern edge?
Why aren't other objects magnified like this? As other objects get farther away, their sizes should also be magnified in the same way..."
Other objects are magnified via looming i.e. any object in the distance that is loomed... eh?Can the effect be enough to account for the 3x magnification that would be necessary for the sun to be the same size when it is overhead and at dawn and dusk at the equator, and the 6x magnification necessary at the southern edge?
Why aren't other objects magnified like this? As other objects get farther away, their sizes should also be magnified in the same way..."
The atmosphere is a plane thus the looming effect is linear in nature and a function of distance from the observer, it accounts for what is observed perfectly.
"...The looming effect raises the apparent elevation of an object, and globe believers use this to explain how we can see the tops of objects that should be below the horizon. How can the flat earth model reject looming in this case but accept it as the reason that the size of the sun doesn't change? Actually, I don't understand how looming is associated with magnification in the first place..."
Looming raises the apparent elevation of an object because the object is enlarged, clearly I'm not rejecting the effect."...How does the flat earth model explain the setting sun? It can't be looming or even refraction in general because those have the opposite effect..."
Your understanding of refractive looming is incorrect, it causes magnification not just an apparent elevation rise. That combined with perspective foreshortening accounts for what is observed at sunset. 
