You ask very profound questions, that I try to answer briefly here, but for a full discussion I would recommend you a good textbook in GR - for instance Carroll's Spacetime and geometry gives a good explanation of the foundations.
In GR objects do not "fall" and there is no force that pulls objects. First keystone of GR: mass(/energy) curves the spacetime in a manner that it is not easy to visualize (maybe impossible in a proper way). Second keystone: objects in their motions follow geodesics.
Geodesics are, roughly speaking, the paths that minimize the distance between points. So the presence of the Earth curves spacetime in such a way that if you pick a point (say, the apple) at some distance from the center or the Earth, the geodesic is a straight line connecting the apple to the center of the planet, hence the motion.
The path that you describe (zero spatial velocity and $c$ time velocity) is not a geodesic for the spacetime deformed by the presence of the Earth, that is why theapple does not follow that motion.
In this sense gravity is not a force: there is no field like in the case of electromagnetism (or weak or strong forces, but these involves other subtleties). In the case of electromagnetism the presence of the charged particle induces a field that can be "seen" by other particles. We say that the particles that are subjected to the field are charged with respect to that field. The particles feel the field because they feel a force connected to it, in the case of EM it is Lorentz force.