Integration?
That is, if the instrument provides a continuous signal proportional to $\frac{dx}{dt}$, you can find the total change in $x$ of $[t_1,t_2]$ from
$$ \Delta x = \int_{t_1}^{t_2} dt \frac{dx}{dt} \approx \sum_{i=0}^n \Delta t \left. \frac{dx}{dt} \right |_i . $$
Of course, as anyone working on inertial guidance knows, that approximation is makes for real difficulty as does the matter of noise and calibration drifts on the instrument. But if you can sanity check the result from time to time (say by comparing with GPS) this will work quite well.
The quick answer is that gyroscopes don't do what you suggest, and indeed don't do anything not expected from the well established laws of mechanics. Because gyrosocopes are so widely used in inertial guidance systems their behaviour has been measured to exquisite precision e.g. in the Gravity Probe B satellite. Any effect of the type you describe would be easily measured.
Having said this, every student is baffled by gyroscopes when they first encounter them, and that certainly includes me. However the equations that describe their motion are very simple. I think the problem is that the gyroscope equivalent of Newton's first law is:
$$ \vec{\tau} = \frac{d\vec{I}}{dt} $$
where $\vec{\tau}$ is the torque and $\vec{I}$ is the angular momentum. The big difference from linear motion is that the torque is given by:
$$ \vec{\tau} = \vec{r} \times \vec{F} $$
where $\vec{F}$ is the force and $\vec{r}$ is the vector from the centre of rotation to the point where the force is applied, and the $\times$ is a cross product. The cross product produces a vector that is at right angles to the two vectors in the product, which means the torque is a vector at right angles to the applied force, and therefore the change in angular momentum is at right angles to the applied force.
It's the fact that gyroscopes respond in a different direction to the force you apply that makes their behaviour seem so counterintuitive. But I must emphasise that it's our intuition that is the problem, not the gyroscope. Once you've mastered the maths, the behaviour of gyroscopes is very simple and entirely predictable.
Response to comment:
The rocket on the left has the gyroscope rotating about it's centre i.e. the gimbal in which it's mounted rotates about the centre. The dotted line shows the thrust provided by the rocket motor. In this case the net torque is zero because the same force is applied to both ends of the gyroscope and the torques cancel out. This rocket would rise normally.
In the right hand drawing the gyroscope is pivoting about it's end. In this case there is a net torque and the rocket would feel a force rotating it out of the plane of the diagram. If the angular momentum of the gyroscope were large enough (it wouldn't be in real rockets) the rocket would indeed rotate round and crash into the ground.
Best Answer
A gyroscope tends to stay fixed relative to it's original direction of spin (angular momentum vector direction).
If a torque is applied it will show some resistance. In some scenarios that torque can be applied by the force of gravity. There are countless scenarios in which other forces can be involved.