Bunch of questions

Hi, Nogueira, and welcome to PF (since it looks like this is only your third post)!

Yes, from the standpoint of General Relativity (GR), the Earth does not feel any "force of gravity" from the Sun, and it moves on the straightest possible path that it can through the spacetime around the Sun. The Earth's path looks curved to us because the spacetime itself is curved by the Sun's mass.

The same way. A satellite orbiting the Earth moves on the straightest possible path that it can through the spacetime around the Earth, just as the Earth does around the Sun. The satellite's path looks curved because the spacetime around the Earth is curved by the Earth's mass.

I'm not sure why you would see a problem with this. The spacetime around the Sun has room for multiple bodies in it. If you're wondering how the deformations due to multiple bodies interact, see below.

Because they have sideways motion as well, just as a satellite orbiting the Earth does. Objects that don't have enough sideways motion, like a rock dropped from a rooftop, do fall into the object at the center (the Earth, in the case of the rock). Put another way, these bodies have angular momentum, and that affects their trajectories.

They do, but which deformations are significant enough to matter depends on the distance scale you are looking at. On the scale of the Solar System, the only significant curvature of spacetime is due to the Sun; the planets are all too small to make much difference. But on the scale of the Earth, the curvature of spacetime due to the Earth is the only one that's significant; the curvature due to the Sun, at the radius of the Earth's orbit, is much too small on the size scale of the Earth.

None of the deformations "reverse" any of the others; they all add up (but which ones are large enough to be significant depends on the distance scale, as above). It's worth noting, though, that the effects due to multiple objects do not, in general, add linearly; GR is a nonlinear theory. (In the Solar System, the nonlinearities are extremely small, so for virtually all purposes we can add effects from multiple bodies linearly and get a good enough approximation; but the exact theory is nonlinear.)

Because their sideways motion is different than that of planets, relative to the size of their orbits. Put another way, their angular momentum relative to their orbital energy is very different from that of planets. The orbital parameters of a particular object don't just depend on the properties of the underlying spacetime; they also depend on the specific motion of the object itself.

 

The apple doesn't have any sideways motion (at least, not enough to be significant), so the straightest possible path for it to follow, in the curved spacetime around the Earth, is the path that free-falls straight down. So as soon as the apple is no longer connected to the tree, it will follow that path. (While it's connected to the tree, the tree is keeping it from falling; and the tree itself is held up by its own structure and by the Earth.)