Hypothetical: Can I determinate if my room is moving by measuring the time it takes for a photon to reach the oposite wall?

You assume that the room is moving close to the speed of light with respect to some absolute frame of reference (this would be the Galilean frame), and thus that the light would appear to you to move slower in one direction than the other.

This is the assumption that was proved false by the Michelson-Morley experiment. That experiment was supposed to dot one of the last 'i's of physics, so that science could close the book on it before the close of the 19th century, and move on to more interesting things. Instead, it absolutely broke the physics of the time. It utterly disproved the existence of a Galilean frame of reference, as well as the luminiferous aether.

To you, light always travels at the same speed. You are always at rest relative to yourself -- it's only relative to other things in the universe that you're moving.

So the whole entire point of Special Relativity is that if the speed of light is constant, then time and space must be variable. So you you and your room are zipping by me at close to the speed of light, it looks to me like your room is foreshortened in the direction of our relative travel, and like your clock is slow -- and it looks to you like I am foreshortened, and that my clock is slow.

All the other stuff in Special Relativity is there to make the math work out, and make it consistent with physical measurement.

No, you cannot tell because the speed of light is the same for you, and in your own reference frame the room is not moving. Remember, there is no absolute motion, so for the observer inside room is the platform that moves (which does not affect the measurements inside the room), not the room.

According to the special theory of relativity, the speed of light in your inertial frame of reference will be the same as in any other inertial frame, so the time it takes for the light pulse to reach the wall on the other side is just $d/c,$ where $d$ is the distance and $c$ is the speed of light.

For a person that you considers being in rest (staying on Earth?), the pulse will travel not only across your room, but also in the direction of your movement. The time it takes according to him is $(d/c)/\sqrt{1-v^2/c^2}$. During this time the pulse travels distance $d$ across the room, and distance $v(d/c)/\sqrt{1-v^2/c^2}$ in the direction of you movement. According to him your clock goes too slow (time dilation).