Speed of light confusion 
I know this question may seem a bit too simple, but I still haven't managed to find an answer to it. 
So, we have two planets (red and green) moving away from each other at a constant velocity (v). At time t1, the green planet releases a beam of light. Say, this light can draw its path in space (so, like a firework). Since light will travel at the speed of light towards the red planet in the red planet's frame of reference, the start of the light's path will seem stationary to the red planet, while the green planet will keep moving away from it. So there will clearly be a distance between the green planet and the start of the light's path.
In the green planet's frame of reference, though, the green planet will be stationary, and so the start of the light's path will stay with the green planet, while the red planet will be moving away.
So, what we get is that the green planet will see that the light starts its path from the green planet, while the red planet will see that the green planet is moving away from the start of the light's path. In other words, one will see that there is distance between the light's path and the green planet, while the other will see that there is not. Of course, this is not possible - because if there was a substance very close to the green planet which would explode if light was shone on it, for one the substance would explode, and for the other it wouldn't.
I know that there is something VERY wrong with this scenario  - it is so basic, and yet there are so many problems with it - but, as long as I have been trying to figure out what, I kept getting to more contradictions.
Can anyone help?
 A: The answer is that you can't mark a path out of pure light. If you want the light pulse to leave a trail like a firework, it needs to travel through a medium which ignites when light passes through it. This makes a big difference because now it matters which of the planets is at rest with respect to the medium.
Suppose the red planet is at rest in this gunpowder filled universe, and green planet is moving. The first scenario happens just like in your drawing; the light pulse will ignite the gunpowder as it passes through; since the green planet is moving away from the point of emission, there will be some space left between it and the firework trail.
Now consider a frame fixed to the green planet. The red planet is moving away and the gunpowder is moving with it. Therefore, the start of the trail, which initially was right next to the green planet, is dragged away with the same velocity as the red planet, so it moves away just like in your first drawing.
The choice of a gunpowder filled universe may seem like I just picked one particular situation in which your experiment doesn't work, but remember the basic point: you can't make a trail out of pure light, because there is no such thing as stationary light. You either make a continuous light beam, which will stick to the green planet, or you fill space with firework material, which will pick out a preferred reference frame.
A: 
Say, this light can draw its path in space

This is where the problem lies. As you have noticed, "its path in space" is not a very meaningful notion. You seem to imagine the photon leaving behind particles that remain stationary, but "stationary" is not a well-defined concept in space.
Something like this is possible in fireworks or in water, say, because there we have an object moving in a medium and the trail can be stationary with respect to the medium. But once you do it in space and you try to implement some mechanism making the trail stationary, you have to specify stationary with respect to what (e.g. with respect to the green or red planet).
