Speed of Dark experiment, does it *really* travel faster than light?
Imaginary things can "travel" faster than light
A shadow or a light spot can seem to travel faster than light, because it's not a particular physical thing, but a series of separate things, separate physical particles emitted at different time and at different locations.
Imagine that you have launched a lot of tiny bots into space with a very accurate clock and a single LED, spaced out in a straight line with a 1 km distance between each of them.
If you program them to blink their LED at particular times – say, the first one blinks at midnight, the second one at midnight + 1 second, the third one at midnight + 2 seconds, then you'd see a spot of light moving at 1 km/s across this line. If you program them to always be on except for a particular moment arranged in the same manner, then you'd see a "shadow" moving at 1 km/s.
If you'd do the same, but set the intervals when your bots light up to 1 millisecond instead, you'd see that the signal is "moving" at 1 000 km/s. If you would have them light up at 1 microsecond difference between the neighboring bots, almost at the same time, then you'd see that the signal is "moving" at 1 000 000 km/s, much larger than the speed of light – but note that there is nothing that's actually moving there, the bots are stationary.
The same applies for true shadows – they're reflected off of something that's not moving (as much), the reflected photons for each moment are different reflected photons, and the fact that a moment ago there was a reflection much further – the "reflection has moved" at above speed of light, describes only the distance and time between two separate events, not an entity that has moved anywhere.
I think you've said a lot of things that are correct, you've just come to the wrong conclusion.
You're right that that it will take us three seconds to SEE the shadow moving across the moon, because that's how long it takes the light to get there and back. But what we'll see is not the shadow slowly moving across the moon over a period of three seconds. What we see is: nothing happens for three seconds (the shadow stays at point A); then, the shadow races across the moon to point B, possibly exceeding the speed of light.
The reason this can happen is simply that shadows are not made out of particles, so there's no reason to think they can't go faster than lightspeed. And as @brucesmitherson pointed out, there's actually no information being transmitted faster than light, since the shadow still takes three seconds to start moving.
It is known that moving a lantern will result in a spot or projected light that travels faster than light is the screen is far enough. However, this effect and similar ones do not refer to the motion of actual objects at faster than light speeds, nor this process allows transfer of information faster than light.