Re: do electrons orbiting an atom's nucleous cause gyroscope inertia?

There are two ways to answer your question. What you've brought up is a question of classical mechanics vs. quantum mechanics. Quantum mechanics was invented to explain questions like yours in a more sensible way around the turn of the century. Niels Bohr's model of the atom essentially put to rest the pre-20th century "planetary" model of the atom. But enough history.

• Do electrons exert any type of gyroscopic force?
  Well, that all depends on how you view the atom. If you say that the atom is like the solar system, the nucleus is the sun and the electrons the planets, all lying in the same plane, then the electron can indeed exhibit the same type of gyroscopic action you may have observed with a toy as a child.
  The electron in this interpretation of the atom's structure orbits, and thus would have a moment of inertia that would give rise to a torque (the actual force that gives gyroscopes their odd behavior). However, if you calculate all the necessary things following a typical college physics text, you would find that using the currently accepted numbers for the mass of the electron and the radius of the electron's orbit, you would find that the moment of inertia is miniscule (well over 20 orders of magnitude smaller than that of a toy gyroscope).

  For example, we know the moment of inertia is 1/2 m r^2, so we plug in the mass and orbit of the electron, and find ridiculously small numbers.

  This gives us a really small effect. Now you might ask, but the electron is spinning so fast, wouldn't it have a large enough torque to measure? Well, yes and no. The kinematical torque (the torque from the electron orbiting the nucleus) is so small, even accounting for the high speed revolution, that we probably couldn't measure it directly (say with some sort of torque-meter). But then, we can't just stick an atom inside a vise or something and twist it and measure it's restoring force anyway- it's simply too small.
  

• The preceeding was of course assuming that the atom consists of the electron fixed in a cicular orbit in one plane around a co-planar nucleus. This is not a realistic model of the atom. In truth, the electron is never "orbiting" in the sence we imagine planets orbiting the sun. The electron does not travel in one plane, rather in a "shell" or "cloud" around the nucleus, with what we would consider to be a practically random orbit.

• So in steps quantum mechanics. The quantity of angular momentum stored in the orbit is a convenient way to characterize the system in a given problem. It turns out that angular momentum is quantized, or comes in discrete packets like an electron's charge. The momentum's quantization has been measured directly through a number of experiments (look for example at the Zeeman effect).

  In the mathematical formulation of quantum mechanics, we take care of the fact that the electron doesn't move in strict planar orbits, we take care of the fact that the electron is so small we can't know its position precisely, etc. This is why quantum mechanics can seem so strange and hard to grasp.

  It is, in fact, due to simple questions like yours and the subsequent study of such problems, that we now can account for the lines in atomic spectra, and can explain the world of the atom in a sophisticated manner.