Spartacus,
It looks like the Gyroscope experts are taking time to get their bearings.
I am not such an expert but I think your answers are as follows:
DGI
2 gimbals, 3 axes, 3 degrees of freedom (including spin) But the spin axis is constrained to remain in the yawing plane of the aircraft.
Attitude Indicator.
2 gimbals, 3 axes, 3 degrees of freedom (including spin), But the spin axis is constrained to remain earth vertical.
Turn Indicator.
1 gimbal, 2 axes, 2 degrees of freedom (including spin).