Thanks for taking the time of going over this with me.
So, gravity is not a newtonian force, the newtonian force is a pseudo force caused by the fact that the earth is a non-inertial frame of reference. It is non-inertial because it bends space-time. Is that correct?
Yes, that is correct. Earth isn't directly a non-inertial frame of reference because it bends space time, but because it is accelerating in relation to other frame of references (which is however caused by bending of space-time).
About your second paragraph, you write "the falling motion is non-inertial", isn't that supposed to be "the falling motion is inertial"? A straight line with constant speeds means no acceleration, which means you're inert? By the way, does this mean if I am free-falling through a vacuum, I am actually observing things from an inertial frame of reference?
Yeah sorry, I meant inertial. Don't know why I wrote non-inertial, seconds before I wrote that I used translate (not native english speaker) to get it correct. But no, you won't be an inertial frame of reference, at least not as an observer. However, your own bending of space time is so negligible that every piece of your body is an inertial frame of reference to every other part of your body, since all of them moves through roughly the same space (and none of them accelerates, neither inertial or in relation to each other, at least no in any meaningful magnitude). A non-inertial frame of reference is one that is not accelerating relative to another frame in the absence of real forces. That is, despite there being no forces, there is acceleration. An inertial frame of reference is one that is completely free of acceleration in relation to another frame, unless there is a force present. So, acceleration only happens when there is a force. The Coriolis effect is an example of a fictitious force, which makes the object subjected to it and the earth (if we assume an object moving over earth) two non-inertial frames. No forces are present, but there is still acceleration. For a car accelerating along the ground, the car and the ground are inertial frames, because they are accelerating in relation to each other, but that is because of a force. Now technically, even in this case both frames are non-inertial ones. That's because of gravity. Since gravity isn't a force, the car is accelerating down in relation to the ground, but no forces are doing it. However, the ground is pushing the car up, accelerating it up at 9.82m/s^2. So the car is subjected to a force, but DOESN'T accelerate. This also makes the ground and the car non-inertial frames of references. This can be applied to everything, so generally speaking, every frame of reference is non-inertial, which is fundamental in general relativity, which gravity is based on (the gravity we are discussing).
What do we call the normal force that the ground is exerting on me while I am standing on it?
It's called a normal force, and it is a newtonian one, so f=ma is present. Weight is what we call the fictitious force which we get if we insert the gravitational acceleration into the f=ma formula. A normal force is any force which acts perpendicular to the surfaces of two colliding objects.