It will not fall over, although it is more difficult to keep upright. You can get bicycle “treadmills” (rollers) which let you ride in place similar to a stationary bike.
No, the energy in the wheels isn't enough to accelerate the bike and your body very much. Either the wheels will spin without traction or you'll lurch a little forward, or some combination of the two.
You can see the effect by lifting your rear wheel off the ground and spinning the tire up then dropping the bike back to the ground. The ground basically just acts like a brake.
Unlikely. You would still be accelerating your mass from zero, and you’d have plenty of time to stop pedaling. If you did it would be an extremely low-speed crash.
I’d say that riding off the side would be pretty likely to cause a loss of control though.
Yes! The gyroscopic force of the bike will completely bypass the braking force of the ground and will be transferred directly onto your body, forcing you to cartwheel like a sideways helicopter into the wall in front of you. Beware! Bewaare!