Is it possible to stay up while riding a bike on a moving sidewalk without actually moving? If I ride a bicycle on a moving sidewalk so that I am not in effect moving at all relative to the ground, will I fall over?
 A: Yes: I've done that. I used to have a device for the purpose, commonly called "rollers". It's like a treadmill for bicycles.
A: What keeps a bicycle up is a variety of things, but it all comes down to the front wheel, which can move left/right. The bike is always out of balance, and if it starts to fall to the left you unconsciously turn to the left, which moves the point of support (the wheel on the surface) to the left, which arrests the fall and may start the bike falling to the right. Then the reverse happens.
You can see this when riding a bike or motorcycle. The front wheel makes small steering motions all the time. An easy way to see this is to try to ride along a straight line marked on the pavement. You cannot do it for very far without turning off to the side.
So it does not depend on your relative motion to the ground outside.
It depends on your relative motion to the surface that supports you, so that you can make these small steering changes to move your point of support left/right.
In fact, there are gadgets you can buy that simply put your bike on rollers and you can ride indoors, as in this video.
A: By the principle of relativity, you will not fall over – assuming that you know how to use the bike and you won't be deliberately "confused".
The principle says that the laws of physics have the same form in all inertial frames that are moving by a constant velocity relatively to each other. The reference frame associated with the moving sidewalk is as good as the reference frame associated with the static sidewalk. In both cases, the bike is moving relatively to it, so if it can stand and move in one situation, it will stand and survive in the other, too.
A: Roughly speaking, a bike doesn't tip over because as soon as you start to tip over, gyroscopic effects (those weird forces you feel when you try to rotate a fast spinning object) together with the structure of the bike cause it steer towards the direction of tipping. Turning in the direction of tipping causes a centrifugal force which will turn the bike upright again (stopping the steering).
A well-built bike will increase the stability as well as a human riding the bike, because it can supply arbitrary forces to balance the bike. (well trained people could balance a bike that isn't moving)
The gyroscopic effects don't require any motion and the centrifugal force only requires that the ground is moving relative to the bike. So yes, it is possible provided that the sidewalk is moving with enough speed. Anyone willing to try at the airport?
A: I would say it is the tire deformation keeps up the bike rather than unconscious steering tweaking; at high speed, high frequency tweaking is impossible. 
Imagine or if you see the cross section of a tire, you can see it is least deformed when the portion is not close to the ground. And it is flattening deformed when it is in contact with ground. The deformation needs force and produce force. There are many factors. But the high the rotating speed, the high the force (due to smaller delta time). The upward force counter-react to any gravity center side shifting and stabilize the bike. 
So whether you ride on moving side walk or anywhere, in order to get you stay up, you need let the wheel spin fast on the surface. 
Sometimes, I was wondering what if a bike doesn't have a rubber tire. 
