Question: Most importantly, what must change in order for the falling object to change its speed? Is it the distance to the centre of the planet? If you pull the earth away from the object as the object falls, will the object slow down or will it keep accelerating?

Answer: 
An object accelerates when a force is acting on the object. This given by the Newton's second law $F=ma$, where $F$ is the **net force** act on the object, $m$ is the **mass** of the object and $a$ is the **acceleration** of the object. The reason why objects accelerate as they fall is because the gravity of earth acts on the object. If you pull the earth away from the object as the object falls, the distance between the object and the earth increases. Now the garavitattional varies inversely to the square of the distance.So, the gravitatational force acting on the object due to earth will decrease and therefore the acceleration of the object will also decrease.

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**Nothing**. As long as there is a **net force** forcing on the object, the object will accelerate.The acceleration will be given by the Newton's second law.
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> Most importantly, what must change in order for the falling object to change its speed?
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Obviously, there's nothing necessary for that. Ignoring ***drag***, the velocity of the object is increasing for sure. Only the acceleration remains constant. Taking *air effects* into account, the object attains the terminal velocity (based on fluid), its $v$ remains constant & $a$ becomes zero. If you're concerned about changing its velocity now, then you either apply another force or *remove* the fluid...

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> Is it the distance to the center of Earth?
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Well, quite... Once the distance to the center increases, the gravitational pull exerted on the object decreases - leading to less acceleration. Hence, velocity changes *indeed*...

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> If you pull the earth away...
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The acceleration on the object decreases slowly. If another body attracts it with some comparable force, your object stops accelerating towards it. But, never quite reaches stationary state...

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Whenever a force is exerted on an object, there's a change in its velocity as a function of time. It's old-school. Check out [Newton's laws](http://en.wikipedia.org/wiki/Newton%27s_laws_of_motion#Newton.27s_second_law) and finally substitute gravity and you're done...