What makes the Earth accelerate in a free-falling object's frame of reference? If I'm an object in free fall near earth, then I'm an inertial frame of reference and I see the earth accelerating towards me with no force acting upon it. What causes that acceleration? The spacetime curvature I cause should be negligible. Is the spacetime curvature caused by earth itself responsible for its acceleration in my frame?
 A: The force acting on the ground is the normal force of the layer of the earth right beneath the ground. The force acting on the layer right below the ground is the normal force from right below that.
Ultimately, however, you can not really understand gravity just by thinking about inertial frame. The whole inertial frame business only works locally: globally, you have to know how spacetime is curved everywhere, and solve the full Einstein equations, etc, to figure out how the earth will behave in the presence of the gravitational field. As I have said, the "force" which acts on each particle of the earth, that makes it move in non geodesic motion, is the pressure (normal force) from its fellow particles.
A: In classical (Newtonian) mechanics, you are not in an inertial frame, so your observations are not valid: the acceleration is fictitious.
In general relativity, the solution is more subtle. You are in an inertial frame, and so is the earth. But in general relativity, inertial frames are not global. The correct way to think about the earth's motion is to determine that it is following a geodesic, and so it has no proper acceleration. The apparent acceleration of earth is purely an artifact of the frame you've chosen, just as it is in the Newtonian case.
You are both in inertial frames, that happen to have a relative acceleration between them. This can't happen with flat spacetime, but there is no contradiction once you introduce curved spacetime.
A: It's important to distinguish between proper acceleration and coordinate acceleration.
An observer in the rest frame of the Earth would see you accelerating with an acceleration of 9.8 m/s$^2$.  At the same time, you would see the earth accelerating at the same rate.  Both of these are examples of coordinate acceleration - each observer labels the position of the other with some numbers, and they see those numbers changing at an increasing rate as time goes on.
However, neither observer is experiencing proper acceleration.  In your reference frame, the fact that the coordinates of the earth are changing at an increasing rate is an artifact of the fact that you have chosen a coordinate system which is moving through curved spacetime.
