My question is: WHY would rotational momentum resist lateral acceleration? As far as I know, an object in motion stays in motion unless disturbed by a net unbalanced force. That implies that the flywheel would be able to apply lateral force against the movement of the robot - and if it does, how does it apply that force?

As a matter of fact - if that were true, surely there would be implications for airplanes (which have large spinning disks as part of their engine), cars (which have a flywheel in the engine), no?

For your specific points:

-If Earth's surface and orbit are not inertial reference frames, what reference frame would cause the robot to "have trouble going upstairs" because it has a flywheel?

-Yes, they flywheel would have mass. However - would the fact that it's spinning change it's translational inertia?

-Rotational IS a part of bipedal locomotion - in one plane. With a single flywheel it is not an issue. So unless the robot is planning to do Olympic-style tilt-starts and tilt-turns when running, that flywheel should not hinder movement as long as it's parallel to the ground.