And how does that let anyone predict transits of Mercury?
Literally all it is, is looking for times the orbit proportions line up; say Mercury being 22.425 days into its orbit and the Earth being 91.3125 days into its, using the occurence of a transit as a zero. Sure, it's a little tricky, but you don't need any Kepler's laws based model, just a few observations of relative position. You'd know how the Sun looks from Earth relative to the stars, you'd know the same for Mercury, using those as landmarks...
The FE analogue would technically be the Sun's path over the Earth rather than the Earth's around the Sun, but the basic principle is still the same.
But my point is how can the transits possibly occur on the flat earth model?
In that model the planets are supposedly at about the same height above the earth as the sun and I fail to see how that geometry could explain these transits.
About the same height does not equal exactly the same height. Only even potential oddity is trying to explain why some sometimes go above and sometimes go beneath, and the rest never transit, but that's a query rather than a contradiction.
Yes, your transit seems to be a sunspot!
But, I will agree on one point. The work of the Islamic mathematicians, astronomers and surveyors of the period say from 600 AD to 1200 AD is largely neglected.
Possibly, but it is still a fair indication that the idea of Kepler being the first is not inherently true. People were watching for similar phenomenon long before, it's not suprising the ones Kepler noted got more notice simply for being predicted, but even so.
Equally, like you say, the work of Islamic scientists of the period is widely neglected. We know they had a fair understanding of lenses and their reaction to light (Ibn al-Haytham, Ibn Sahl for the easy examples; the latter of which discovering Snell's law centuries before, well, Snellius and his contemporaries, and the former doing plenty with the magnifying properties of lenses), it's wholly possibly Averroes didn't use his naked eye given they were a century before his time. A lot of the records are missing.
In Heliocentric system Kepler's laws are applied on Earth as well.
On what is that part applied in Flat model where Earth is unmovable?
They're not. Why is that relevant? No one's claiming the models are identical, just that regular, repeating phenomenon are predictable.
Jane's not 100% wrong. Planetary motion was accurately observed and modeled long before we had any mathematical understanding of what was happening. Ptolemy's geocentric model is functional to describe what is observed in the sky, if you allow a margin of error and don't mind planetary epicycles that exist for no explainable reason other than that's the apparent motion that was observed.
The Copernican model is also based on observation, but it has much more consistent planetary movement and the engine of gravity to explain how it all works. The better our mathematical descriptions of gravity have become the more accurate the Copernican model has become.
Jane's mistake here is trying to allege that the same observations will be consistent if the Earth were flat. That's absolutely false, but Jack explained it above more succinctly than I would, so I won't bother going through it again.
The observations are consistent. Observations aren't going to change. What the hell they mean, meanwhile, is entirely different. Even if you want to take the most extreme potential distortion, you're going to get stars teleporting across the sky or two identical things existing simultaneously etc etc, which is all a hell of a job to explain, but (barring entirely separate arguments against FET which can and so should be made independently of astronomical software) it's all still perfectly predictable.
I've gto Jack blocked, though I did skim his post when you mentioned it, and 'succinct' is not the same as 'rapidfire.' It's the reason I've got him blocked; he is never trying to make informed arguments, just cheap quips. When you cut parts of my post out of context, sure, you can quip a one-sentence response that seems to refute it, but you need to actually ignore everything that's being said and refuse any attempt to actually understand in order to really do so.
Why things happen, and what happens, are not the same thing. Trying to explain why the things we see in the sky happen under FET is a long, complicated task, but knowing that it's going to happen isn't the same.