If I have a charge of $+q$ placed arbitrarily within the spherical conducting shell, by Gauss' law, the $E$ field produced will be created outside the shell, as if the charge were placed at the center. If this charge is not placed in the center, I don't see how we can use symmetry to argue why the outside should not be able to 'see' how the inside of the shell looks like.
I'm aware that the charges in the conductor rearranges to make the Electric field within zero, but it is not apparent to me that this rearrangement should 'balance' out any perturbations created by the moving the enclosed charge off the center.
Best Answer
Here is my attempt to explain the final E-field pattern.
A point charge $+q$ produces radial E-field lines.
Diagram 1
Suppose that a conducting spherical shell is placed around the point charge with the point charge not at the centre of the conducting shell.
Since the E-field inside the conductor must be zero there must be another E-field inside the conductor (red) which is equal and opposite to that produced by the charge $+q$.
This induced field is produced by induced charges formed on the inside and the outside of the conducting shell.
For a number of reasons this cannot be the final state of the system one of them being that E-field lines must be perpendicular to the surface of a conductor.
Diagram 2
The E-field lines are now all perpendicular to the conducting shell.
As soon as this condition is satisfied (some) information about the position of the charge $+q$ when viewed from outside the conducting shell is lost because all the field near the outside surface of the conductor appear to come from the centre of the conducting shell. This diagram is again flawed one reason being that in this configuration the conductor is not an equipotential.
The work done in bringing a positive charge from infinity to the left hand surface surface of the conductor is more than the work done in bringing the same charge to a point on the right hand side of the conductor.
This cannot be so because the conducting shell is an equipotential.
Diagram 3
The induced charges on the outside (and inside?) rearrange themselves so that the work done in bringing a change to any point on the outside surface of the conducting shell is the same whilst at the same time making sure there is no E-field inside the conductor.
So to the outside world outside the conducting sphere the actual position of the charge $+q$ is unknown and the E-field is the same as that which would be produced by having charge $+q$ at the centre of the spherical conducting shell.