Aah. A common confusion when dealing with falling elevators.
The base point is, if you are in freefall, you will not feel any force; you will only see your surroundings moving fast. If you are in a closed elevator in freefall (neglecting viscosity and the other stuff), you will feel weightless, and there will be nothing to indicate that you are on Earth in freefall.
The main difference between a rocket and a falling elevator is that gravitational force is not acting (at sufficient distances from the Earth). Whenever a 'rocket' is used in any explanation of relativity, assume that it is far from any gravitational bodies.
I shall refer to psuedoforce from here onwards, please see my brief explanation below if you do not know what it is
In the rocket, you feel a pseudoforce ($=-m\vec{a}$) opposite to the direction of acceleration, which presses you to the wall/floor/whatever.
In a falling elevator, you also feel a psuedoforce ($=-m\vec{g}$), but you are also being attracted by Earth's gravity ($=+m\vec{g}$). So the net force you feel is zero.
Another way to look at it is this way (in ground frame): In a rocket, your inertia wants to keep you at rest, but the rocket is moving. So, when the rocket starts (in gravity free space), you will stay stationary with respect to space, but will move with respect to the rocket. Once you hit the wall of the rocket, the rocket can "push" you along so that you travel with the same acceleration as the rocket.
On the other hand, in a falling elevator, the elevator is moving with $g$, and so are you (from the ground frame). So there is no need for that extra "push" from the elevator.
Summing up
- "Why can't we think of the elevator as upside down rocket?": This is because gravity is acting upon the elevator, but not the rocket. The elevator is being accelerated by gravity (so are you), but the rocket is accelerated by its engines (and you only get accelerated if you stick to a wall and are pushed)
- One feels weightless in freefall
Oh, and one more thing, there is no meaning for "goes up" and "goes down" for a rocket in space. Its ok to use this while referring to a single diagram, but as there is no "up" and "down" to space, the first two diagrams are exactly the same thing. It's the way they're drawn which has led to the confusion in the first place.
Psuedoforces
Whenever we view a system from an accelerated frame, there is a "psuedoforce" or "false force" which appears to act on the bodies. Note that this force is not actually a force, more of something which appears to be acting. A mathematical trick, if you will.
Let's take a simple case. You are accelerating with $\vec{a}$ in space, and you see a little ball floating around. This is in a perfect vacuum, with no electric/magnetic/gravitational/etc fields. So, the ball does not accelerate.
But, from your point of view, the ball accelerates with an acceleration $-\vec{a}$, backwards relative to you. Now you know that the space is free of any fields, yet you see the particle accelerating. You can either deduce from this that you are accelerating, or you can decide that there is some unknown force, $-m\vec{a}$, acting on the ball. This force is the psuedoforce. It mathematically enables us to look at the world from the point of view of an accelerated frame, and derive equations of motion with all values relative to that frame. Many times, solving things from the ground frame get icky, so we use this. But let me stress once again, it is not a real force.
