I found a website run by an interesting fellow called 'Wild Heretic' and it seems incredibly intricate and comprehensive. I've yet to see any other person argue as well for half so radical a claim. Think of this as an opportunity to examine arguments for highly unpopular views.
Wild Heretic believes that we live on the inside of a hollow sphere, lit by a half-light half-dark Sun at its center (he claims that light bends in order to produce the effect of rising and setting), that the moon is an optical illusion, that manmade satellites don't really exist, that the stars are light artifacts produced in the atmosphere and can never be seen above it, and he has a bunch of explanations for the other celestial bodies like comets and galaxies.
It all seems shockingly intelligent (aside from when he insists that the fact that the Earth doesn't move under your feet when you jump disproves heliocentrism). He also has nine main pieces of evidence for his model:
1. Some early modern maps have inversed latitude and longitude
2. Modern polyconic maps show more accurate sizes and shapes
3. 19th century balloon observations (that is, without an intervening medium) gave the impression of a concave surface
4. 4,000 foot plumb lines reportedly were farther away from each other at the bottom of a mine shaft
5. A laser shot between two posts (over water) seems to curve downwards
6. An old rectilineator experiment indicates a concave surface (the experiment has been criticized here)
7. Radar and radio wave horizons cannot be explained on a convex ball
8. Ships disappearing below the horizon are an optical illusion
9. Light bends upwards, which allows for the rising/setting illusion of the sun and moon
I would really like to know what people here have to say about this, since the comments on the site itself are very disappointing. (A lot of it does rely on a massive conspiracy involving scientists of many stripes, but it's probably best to overlook that.)
I don't have time to refute each of arguments, because there're too many. But consider number 5 in your list. He describes a laser experiment that he claims cannot be accounted for on the current picture of the Earth. But if you think it through, it is perfectly well accounted for.
Here's the version of the experiment performed by the two Polish guys on a lake. They place two stakes 2km apart. The stakes have lasers attached to them at 30 cm height from the surface of the water. They measure the height above the surface of the point at which the laser beams meet and find it to be 39-40cm above the surface of the water.
Wild Heretic claims, on the basis of this diagram, that on the convex Earth theory (i.e., the widely accepted theory) one should expect the height from the water at the point where the lasers meet to be smaller that the height at which the lasers are mounted. But Wild Heretic's diagram misrepresents the state of affairs. Here is a better representation I drew and associated calculations that I did, which show that the convex Earth theory correctly predicts that the laser beams would meet approximately 38cm above the surface, which is very close to the observed 39-40cm.
EDIT: As dogiv points out below, I mis-interpreted the experiment. So the argument above is not a refutation of the experiment as described.
This doesn't actually seem to match the description. They only talk about having used one laser, with two stakes, whereas your diagram requires using two lasers. Your setup would be quite difficult to achieve, since you would somehow have to get both lasers perfectly horizontal; I'm not sure a standard laser level would give you this kind of precision. In the version they describe, they level the laser by checking the height of the beam on a second stake. This seems relatively easy.
My guess is they just never did the experiment, or they lied about the result. But it would be kind of interesting to repeat it sometime.