Why do objects "fall" along spacetime geodesic lines? I'm working on a paper that also addresses the topic of general relativity (among other topics).
The most common answer I get to the question above (why do objects fall) is that the objects are not really stationary, they are moving through spacetime, and thus they are forced to follow spacetime geodesic lines. I have a hard time understanding this explanation... How is an apple moving through spacetime? When looking at the spacetime continuum from the outside, isn't the "moving" 3d apple actually a stationary 4d object?
Even more, in the common representation of the Earth bending the fabric of spacetime, I understand that the 2d grid is meant to be a representation of the 4d spacetime continuum. Then why is the Earth represented as a sphere? Shouldn't it be represented as a (long) cylinder instead, so it's existence in time is made clear?
Next, shouldn't the falling apple be represented as a much thinner cylinder, which at some point (when the stem breaks) starts getting closer to the larger cylinder (the Earth)?
 A: First, only a test particle "falls" along a geodesic. A test particle is an idealized object not only at rest, but which also does not itself contribute to the curvature of spacetime (no mass, no energy). An apple can be considered as a test particle in a system including earth, but it is a simplification, as the overall spacetime is determined by the dynamics of all objects that supposedly live "in" it; see this related answer of mine. 
Now a massive object would also follow a geodesic, if we take this object into account in the spacetime itself by considering how it itself distorts spacetime. See this question - as John Rennie says there, it is a matter of terminology. The main point I want to make here is that spacetime is not a background. There is no "fabric" of spacetime.
Second, and as you correctly state in the question, a geodesic is not a purely spatial trajectory, it is a 4-dimensional curve, so it is actually misleading to think about "falling along" a geodesic, because the dynamical aspect of "falling" is already a part of the geodesic itself. A geodesic represents the world line of a test particle, and as such it is static: it tracks the object position in the most generalized sense, in all possible observer-relative decompositions in space and time of the unified spacetime it lives on. In other words it says where the particle is at any time in a way that is independent of any observer, an absolute way. 
This means that "falling" along a geodesic is simply equivalent to "being somewhere" for a relativistic object. There is nothing forcing an object to follow a geodesic, a geodesic just tells when/where an object is, for its whole existence, an existence during which nothing ever messes with its position/momentum (nothing except gravity, but precisely because gravity is replaced by geometry in general reativity, it amounts to a non-interaction; see the answers to this question for a related discussion of the equivalence principle).
In short, a (timelike) geodesic is a geometric statement about inertia: where are you when you do nothing and nothing does anything to you? On a geodesic. 
A: In a curved manifold, as spacetime in GR (general relativity), a geodesic is a curve followed by a non interacting (free falling) particle, whether massive or massless (photon). It is the extension of the straight line concept of SR (special relativity) as in Minkowski flat spacetime.
To figure out why objects are described as falling, let us consider the cartesian coordinates in Minkowski spacetime. Even if a massive particle is at rest, its time coordinate does not stop, hence the path is a straight line, parallel to the time axis. If the particle is moving with a uniform speed the straight line will present an inclination. In a curved spacetime the worldline of a particle has a less simple shape, but it describes a path anyway.
As for the other points related to the pictorial representation of the fabric of spacetime, probably it is an artistic view.
