The Earth is flat

Flat Earth Theory is easy to justify with three basic observations. 

1. The Earth is either round, or it is flat. 
This just comes from observation of nature. Certain shapes are preferred by natural forces, from the spherical shape of bubbles to the flat surface of water. Yes there are minor perturbations, but objects will tend towards least surface area or least irregularity; for example, a flat surface would arise when any two forces meet, along the plane of impact. 

2. ""If I have seen further it is by standing on the shoulders of Giants." -Isaac Newton 
Science is based on what came before. Certain things become so entrenched they are simply not questioned any more. At best you have half-hearted handwaves to placate the masses with no degree of rigor. Space travel was supposedly invented during the cold war, the propaganda of the US and Russia plainly cannot be trusted at that time. Since then were it a lie neither country would want to admit it; thus, scientifically, we need some actual way to choose between the two possibilities. What we get is suspect. For example, one popularized notion is that of reflectors on the moon's surface, the lunar laser ranger experiments, in which reflectors were placed on the moon during the Apollo missions. This is meant to indicate we must have been there. However we have no indication that this phenomenon is not natural, there are no records of lasers being directed at the same parts of the moon's surface before the Apollo missions. As such this fails to be evidence of anything actually being put there. 
Science has become an appeal to tradition, not to logic. 

3. Gravity is a flawed hypothesis
It is not understood in the slightest. A myriad claims are needed to be made about it to have a hope, culminating in the supposition of dark matter to explain celestial movements. This truly is the death knell, as if the majority of the mass of the universe was dark matter, how was it not noticed sooner? How did we not see that the mass of the Earth (as dark matter a) obeys gravity so should be attracted to the same locations as normal matter, and is meant to be present where we are) was several times that which its composition allows for? How has there not been a major overhaul of every single scientific discipline? 
When gravity is rejected as the unscientific dross that it is, the least surface area of the Earth may be rejected as the least likely. 

This is without touching issues like Occam's Razor, water levels, the traits of a force that keep us on the Earth's surface, etc etc. All of which directly favor a flat Earth. 

It is amusingly suspect that you choose to link to the FES as the bastion of Flat Earth Theory. It is swarming with round earthers, almost every active user on the site is, including the moderators, and the view advocated quotes Leo Ferrari, a satirist, as a scientist. Try IFERS. 

I've no doubt the science more or less checks out, it's just based on false presuppositions.
The claim that a random rocket reaches space is false.

It doesn't. It goes up. Then it goes down somewhere else. It doesn't go into space any more than any random plane I could mention.

The near-vacuum about 100km up, I doubt anything can get close. I'm skeptical of anything over 50km, not that it's impossible, just not too easy. The vicinity of 80/90km is my limit.

You asked me to define space. The karman line isn't the bit I have a problem with, 80-90km is the range we observe meteors, it's where chunks of rock and metal that fall to Earth are torn apart by sheer friction.
Depends what you're calling the ISS. I believe that light we see in the sky exists, I just don't find it any more remarkable than a typical plane, and I doubt it's anywhere near that high.

Do you have an account of that being done? It's pointless speculation if not.

I'm well aware of what trigonometry is, but giving a claim doesn't make it reality. Yes, that experiment could theoretically work, but it's useless to bring up unless you have figures to offer.
To use the easiest example:



This diagram comes form your second link. I am using it as it is easily the most explicit. You can calculate the height first by calculating the diagonal distance to the ISS via the sine rule (using the fact the sum of the angles in a triangle sum to 180), and second by the simple definition of sin. This does not give a height of 287, it gives a height of 343, substantially off from even the RE figure. You are welcome to try. If I am missing something, please do inform me, but it seems that these measurements are even by your own model bs. 
The sine rule gives us the hypotenuse of any triangle where the height of the ISS is the opposite side to the given angle. Given that the sin of the known angle is o/h, it's trivial to find o. 

Why is this?
First of all, these figures are not accurate. Error is always a factor. There are factors with timing it is harder to account for in this (synchronising the measurements across both locations perfectly etc), so we would expect a departure already. 
Second is the distance between the two locations. You cannot reliably measure 370 miles, both in terms of measuring the distance and determining relative locations with that much accuracy. Let's suppose it's in fact closer to 375 miles; wholly reasonable. 
And then the angles. Measuring the azimuth of a star is not easy, at all. Even measuring angle with respect to a truly level surface cannot easily be done to within 0.05 degrees, and to go from that to measuring stars without clear landmarks can easily net you, say, 4 degrees of error, at least. 

If you switch links to your first link, you see that they correct for potential angle error to the lower bound figure I gave above, and a figure in the ballpark of 250 miles suddenly becomes less than 90 miles, and that's just the one factor. No mention of distance, timing...