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Yes, they feel the same, and this observation is fundamental to how we think of gravity. Einstein said that not only do they feel the same, they are the same: movement under gravity alone is the same thing as movement under no force at all. The name for this assumption is the equivalence principle, and it underlies General Relativity: because we know that things experiencing no force at all move in straight lines through spacetime, we also know that things moving under gravity alone move in straight lines through spacetime, and this works because what gravity does is to curve spacetime, so that 'straight lines', which are now called geodesics, have properties which straight lines in a flat spacetime do not have, such as intersecting more than once.

To be slightly more precise about this: there is (in GR) no local distinction between movement under gravity alone and movement under no force at all: because gravity distorts (curves) spacetime, there are experiments you can do which are not local which will tell you whether you are moving under gravity or under no force. Geometrically, these experiments consist of establishing whether straight lines have the properties you would expect in a flat spacetime or whether they have properties you would expect in a curved spacetime; physically the experiments consist of detecting 'tidal forces' which are forces which cause two separated objects (the being separated is what makes the experiment non-local), initially at rest relative to each other, to want to move away or towards each other over time.

In essence, yes. Being on a space station in orbit basically IS falling due to gravity, it's just that the astronaut and the space station keep missing the Earth due to constantly moving sideways so they never hit the/fall on the Earth. But they basically ARE falling.

Our bodies can't tell the difference, because all your body parts are accelerating and moving at the same rate, they're not in any tension in relation to each other so it's like there's no force, none that you, the person, can feel anyway.

There are some minor differences, tidal forces, but these effects are minor unless you're orbiting near a black hole etc. Tidal forces: slightly stronger gravity near the gravity source, so your feet, for example, are pulled sightly stronger, but these effects are minor usually. Astronauts on the ISS certainly don't feel it.

The term "zero-g" just means you don't feel any gravity, not that there isn't any. Of course, if you were in the void, far far far away from any gravity source, you would still be in "zero-g" because you wouldn't feel any... because there is none.

"g" here refers to a thing called "gravitational acceleration on Earth" btw, which is $g=9.81:rm m/s^2$. Fighter pilots go through 5g and more because they accelerate a lot... gravitation itself being irrelevant here, it's all about the felt acceleration itself. Emphasis on felt. Astronauts accelerate too, as I've said, but they, the persons, don't feel it, because they aren't squished onto anything, like the fighter pilots are squished onto their jet engines.

This answer mainly expands on earlier ones as I think a little more can be said on tidal forces.

Floating in space and falling under a uniform gravity are indistinguishable if you don't have any external reference points to observe. However, if you are falling feet first (for example) towards Earth, or any other planet, then gravity is not uniform for a couple of reasons.

Firstly, your feet are slightly closer to the centre of the Earth than your head so your feet experience slightly stronger gravity than your head. This is experienced as a (very small) force trying to stretch you from head to foot.

Secondly, because the attraction is effectively towards a single point at the centre of the Earth, the direction of gravity is very slightly different for your left shoulder and your right shoulder. This leads to a very small net force compressing you from each side of your body and front to back as well for the same reason.

In practice, with something as small as a human and such a comparatively weak gravity, you won't be able to detect the differences but these are the same forces which generate tides when you get to the scale of the Earth & Moon. Going further, Stephen Hawking came up with the word spaghettification in "A Brief History of Time" to describe the effect of an object getting too close to a black hole and experiencing these forces. The name says it all, really.

Yes, they are both same (with at least one exception given below), because their state (of motion or rest) is only being influenced by "curvature of space" alone. There is no other external force at work.
Because they are freely moving/floating under influenced by "curvature of space", they do not feel that curvature. That state is referred to as weightlessness. They both feel weightless.

There is one exception though - near the black hole, the spaghettification becomes noticeable/observable/painful.

So, someone falling freely near a black hole, will have different feeling as compared to someone floating freely into far space, or falling freely around an ordinary planet.

While the physics is equivalent, the two sensations might well be perceived of as different. The system sensing accelerations tends to interpret higher frequencies as translational motion, and lower frequencies as a reorientation with respect to normal gravity. (See Seidman, S., Telford, L. & Paige, G. Exp Brain Res (1998) 119: 307. https://doi.org/10.1007/s002210050346), for example). People rarely fall forever. Sometimes we fall for a long time, though. I would imagine that the sensory experience of space might approach that of a parachute jump, for example, which would have very low frequency components.

Also, our sensory systems know that we live in a 1-g environment. There are a number or famous illusions that occur when this is violated (Cohen, Malcolm M. "Elevator illusion: Influences of otolith organ activity and neck proprioception." Perception & Psychophysics 14.3 (1973): 401-406, for example)