One question. Why put the flywheel close to the head? (Also I don't think flywheels work that way - most likely it would just look strange, having an ultrastable head - maybe, if built as to make this, have a slouched walk as nothing above the hips except the head is active, resulting in a two-point stabilization - the legs and hips, from the ground, and the gyro head.

Cars have a (by my math) 4kg flywheel spinning at, usually, 2000 to 6000 RPM. Planes have (by my estimates - I don't know the exact materials used or exact volume of moving parts, but for a JT9D - the engine used in a lot of 747's, I'm estimating about 25% of moving mass, and the engine weighs 3905 kg - so that alone would be close to 1000kg, so I'm generalizing) upwards of 500 kg of moving parts spinning at upwards of 10 000 RPM. That's a lot of energy, and far from insignificant. How insignificant? The ISS, a space building, uses only a few flywheel systems no larger than a man.

-If Earth's surface and orbit are not inertial frames, then what would be the reference frame in which one would have trouble carrying a moving flywheel upwards? Also, 12kmiles - that's about half the circumference of Earth. Let's assume, for pure terror factor, that this machine can run at about 30 mph - faster than the fastest human sprint speed recorded - constantly. To run 12k miles, it would take it about 400 hours. That's a correction of about half a degree an hour - I'm pretty sure that unless you had a supermassive flywheel, the body would have no trouble applying the force necessary to do it. (And if it didn't - we can use gyroscopic precession to apply force to the flywheel to force it to tilt). It most certainly would not fall over, though.

-The flywheel shouldn't be more than 10% (if that) of the mass of the machine. It's not especially significant. We humans deal with worse regularly.

-Walking and running doesn't create as much torque as you believe. We don't lean forwards just to walk - and we're not doing it that significantly while running. A flywheel would remove the need for that. Tilt-turns maybe - but as previously mentioned, you can apply force to the flywheel directly to make it tilt OR you can give the legs a wide angle of operation and the issue goes away.