The reel's outer circumference is twice its spindle's circumference, so to unspool 100 length of cable it will roll a distance of 200. With the elbow room, you need the hallway to be 210 long. To keep holding the end of the cable during the unspooling, you will have to walk forward half the speed of the reel, travelling about half the length of the hallway.
Just pull the cable, walking back to the start of the hallway. The cable was being pulled forward during unrolling, since it was being unspooled at half the speed of the reel. So by pulling it back, the process reverses, with the reel moving towards you at twice the speed at which you pull the cable. When you reach the start of the hallway, the reel reaches you and the situation is as it was at the beginning.
The bike moved back. The pedal is like the reel's inner spindle in Q2. Imagine you standing behind the bike with a cable attached to the bottom pedal, pulling the cable. Just as with the reel, the cable unspools slower than the bike itself (but even more so due to the gearing) so you pull the bike towards you.
You need the pedal to move back relative to the bike faster than this moves the bike relative to the ground. So the pedal needs to move back faster than the bottom of the wheel moves back, both relative to bike. This needs A/B < D/C. One way to make this happen is to swap the two gear wheels, swapping the values of C and D.
It may seem strange that the bike can move in the opposite direction to the push applied to the pedal. One way to explain it is as follows. Imagine the pedal being fixed onto the rear wheel itself, without gearing. If the pedal is at the bottom, then the spoke going from the hub of the wheel past the pedal to the bottom of the wheel is like a lever. The point of the wheel on the ground is the fulcrum, and when you push the pedal which is halfway up the lever, the other end of the lever pushes the bike faster in the same direction. When you introduce gearing, you change how much effect the movement of the pedal has. Normally you add higher gears, which is like moving the pedal up, closer to the hub so that less pedal movement leads to more bike movement.
In this case we add an extremely low gear, which is equivalent to moving the pedal down so far that the spoke that acts as the lever would have to be extended below ground. It is obvious that pushing one end of a lever makes the end on the other side of the fulcrum go in the opposite direction. In this case of the bike it is a bit counter-intuitive that the gearing allows the same thing to happen without actually needing to be on the other side of the fulcrum.