You can learn a lot by experimenting with a bike wheel. Hold each end of the shaft in one hand and roll it along the ground in front of you to get it spinning. You will find that if you then try and turn the wheel (like steering a bike) that it tries to lean over (eg you get a force at 90 degrees to that which you apply).
Going back to the question... (Does the axis of rotation = axis of sensitivity or not)... I think it depends if your giro is a mechanical type (like your bike wheel) of a fancy laser type....
If it's mechanical then I believe the axis of sensitivity is at 90 degrees to the axis of rotation as demonstrated by the bike wheel.
If it's a laser type then I believe the axis of sensitivity is the same as the axis of 'rotation'. See the section on 'Ring Laser Gyros' in this long article...
http://www.repairfaq.org/sam/laserlia.htm#liarlg
For info... I have in my hand a single axis piezo based gyro that complete with electronics and cables weighs approx 8 grams.
Similar to this one..
http://www.hobbico.com/accys/hcam4000.html