Lets tackle each argument one at a time...
When a policeman fires his 'speed tracking instrument' at a car the light produced reflects from the car and according to the speed of the car frequency changes to an apparent observed frequency. (Doppler Effect)
So we could just fire up a light beam to a wall and detect the reflected light. If it has another frequency we detect motion of the box.
What you say about the policeman firing a radar gun at a car is entirely correct: the Doppler Effect will certainly have an effect in this situation and thus show that the car is moving. However where the problem in your translation of this argument to the situation of a moving box lies is within the idea of different Frames of References. In the car situation, the car and the policeman are in two different frames of reference, given that the policeman is stationary and the car is in motion. Therefore in the policeman's frame of reference the car is moving, and this relative motion will give rise to the Doppler Effect. In the moving box situation, however, the person and the box are in the same frame of reference. Therefore when the ray is fired off from the radar gun, it will be moving at the speed of light relative to both the person and the box. As a consequence, there will be no Doppler Effect and hence it is not possible to observe that this frame of reference is moving.
Now imagine that there are windows on this box which is flying through space at a constant velocity. If the person inside were to look out of them, he would see the Doppler Effect acting on all objects outside his box. Why? Because these objects are in a different frame of reference. However it is important to realise that this would still not tell him if he was moving - it could very well be the objects outside moving rather than him, for which the effects would be the same. It is therefore impossible for him to know if he is in motion at a constant velocity.
The box moves around earth with us on it. So the centripetal acceleration must be provided by the gravitation pull of the earth as no other force acts downward. This Force must be greater than the normal force for it provide centripetal force which is otherwise equal to the normal force if we weren't moving.
First of all, there is no normal force - the only force acting here is the force of gravity towards the center of gravity of the planet. In regards to relativity, you are in this instance in a non-inertial frame of reference: you are moving in a circle, which means you must be accelerating, a fact which you yourself pointed out, saying that "the centripetal acceleration must be provided by the gravitational pull of the earth." In this case the postulates of Special Relativity no longer hold, so you cannot use this argument to disprove it.
Suppose a gigantic movable wall moving in air then intuitively the pressure at the side of the wall in front (that is facing the air molecules) to have a little bit more pressure than the other side (behind the wall).
Similarly we could measure the pressure at points close to the 4 walls. I expect the pressure a little less on one side then the other.
This argument has more or less the same flaw as the previous: moving through the air will exert a force on the wall (hence the presence of the pressure you mentioned) and as a result you are no longer in an inertial frame of reference - you will experience an acceleration due to this force. Hence Special Relativity does not apply here, and thus this argument also cannot break the postulates of SR.
Hope this was helpful. (If there are any mistakes, please point them out in the comments.)