Please check out [url=]Magnetic field pattern for (line/ coil/ solenoid)[/url] for magnetic field line pattern around solenoid.
The magnetic will have the same displacement when you move the magnet (the magnetic field line pattern will not be the same if you can move magnet near speed of light-- which is impossible).

The induced voltage will proportional to the magnetic flux change
V=\frac{d\phi}{dt}=\frac{d}{dt}\int\vec{B}\cdot d\vec{s}

The deflection of galvanometer needle is proportional to the voltage.
Increasing the speed of magnet will increase the rate of flux change.
I will let you think about other questions you have raised. ;)