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: 
As long as there's a non-zero net force acting on the object, it will have a non-zero acceleration and therefore it will continuously change its velocity: $$\vec{F} = m\vec{a}.$$

In the case of gravity, the force is inversely proportional to the distance between the objects squared: $$\vec{F}\_G = G\frac{m\_1m\_2}{r^2}\frac{\vec{r}}{r},$$ where $\vec{r}$ is the vector connecting the two objects and $r=|\vec{r}|$ its length. So the closer the objects are, the stronger the force or -equivalently- the acceleration. Notice that the acceleration is only zero if the objects are infinitely far apart. (I'm assuming no drag, let's only consider the gravitational force for simplicity)

If you pull the earth in the same direction the object is falling so that you maintain the same distance $r$ at all times the object will just keep falling with a constant acceleration. If you pull it faster, the distance will increase and the acceleration will therefore decrease, meaning the velocity of the object will increase more slowly than before. But it will never decrease. Pulling the earth more slowly will only decrease how much the acceleration would increase if you hadn't pulled, so again the velocity keeps increasing.

So to summarize, the object's velocity will always increase, unless you can get the distance to infinity, which should only take you about - an infinite amount of time. And even then you can only get to a constant velocity, never a decreasing one. You need a repulsive force for that (or additional attractive forces on the other side of the object).
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.