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Travelling faster than the speed of light

Please try to follow and clarify me I've been curious about this for a long time as it seems a bit paradox due to the rule that you can't pass the speed of light.

1- Lets consider we are in a room with closed windows. When we turn a lamp in it's centre it shoots out light in all directions. So the light hits the closed windows at the speed of light.

2- Now lets consider that one of this windows is turned to the north, and we open this window. It makes sense that the light now exits the room through this window and we see it moving away from inside the room at the speed of light.

3- For the last step let's now consider that the room is moving north at the speed of light. What happens to the light that exits the window?

3.1- From inside the room I would view the light moving north at the speed of light

3.2- From outside the room, observing from a stationary position, would the light exiting the window move at twice the speed of light? or would it change speed?

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From inside the room I would view the light moving north at the speed of light

That's true.

From outside the room, observing from a stationary position, would the light exiting the window move at twice the speed of light? or would it change speed?

No, it will still move at speed of light, thus theoretically outer observer will never see any light coming out of that window in this case, this is simple result of Velocity-addition formula in special relativity, check the link.

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