Is it possible for information to be transmitted faster than light by using a rigid pole?
The answer is no. The pole would bend/wobble and the effect at the other end would still be delayed.
The reason is that the force which binds the atoms of the pole together - the Electro-Magnetic force - needs to be transmitted from one end of the pole to the other. The transmitter of the EM-force is light, and thus the signal cannot travel faster than the speed of light; instead the pole will bend, because the close end will have moved, and the far end will not yet have received intelligence of the move.
EDIT: A simpler reason.
In order to move the whole pole, you need to move every atom of the pole.
You might like to think of atoms as next door neighbours If one of them decides to move, he sends out a messenger to all his closest neighbours telling them he is moving. Then they all decide to move as well, so they each send out messengers to to their closest neighbours to let them know they are moving; and so it continues, until the message to move has travelled all the way to the end. No atom will move until he has received the message to do so, and the message won't travel any faster than all the messengers can run; and the messengers can't run faster than the speed of light.
/B2S
The information about the pushes will be received on the other end with the speed of sound in the substance of the pole. For any real material it is much slower than the speed of light (for a steel rod it would be about 5000 m/s).
No.
In relativity you cannot consider extended objects to be infinitely "stiff" - they must bend and stretch, as real objects do. When you move one end of the steel rod, it makes part of it bend and stretch which exerts a force on the next section which makes that move and which makes a new part bend and stretch and so on and so on until you reach Alpha Centauri. This moves along at some speed which is characteristic for the metal which is fast enough that we don't really notice in day to day life. All relativity tells us is that that characteristic speed is less than the speed of light - it turns out for real metal its much less than the speed of light.