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I'll address question #1 first. I think this one has to do with the fact that Cooper and TARS actually reached one of the singularities inside the black hole, where TARS was able to gather the needed "quantum data" (my answer here covers why this was necessary), and that was the point where they were rescued from the black hole by the "tesseract" which had been created by the beings living in the higher spatial dimension. This bit of dialogue between Cooper and Romilly on Mann's planet is key:

The "tesseract", meanwhile, is supposed to be a piece of technology created by the beings (possibly descended from humans) who live in the extra spatial dimension, the "bulk". This idea of an extra extended spatial dimension is based on a real physics theory, the Randall-Sundrum model--see my discussion in this answer for more details. The tesseract is shaped like a four-dimensional hypercube (that's what the word 'tesseract' means, in fact), so each of its "faces" is a 3D cube, just like each face of a 3D cube is a 2D square. In ch. 29 Thorne describes how the tesseract can "dock" one of its faces to our ordinary 3D space, which in the Randall-Sundrum theory is a 3D brane sitting in the 4D space of the bulk (for anyone familiar with the classic "math fiction" story Flatland, I think this docking of a higher-dimensional structure with our space is meant to be analogous to how the 3D sphere was able to materialize in the 2D universe by having one of its cross-sections in the 2D plane). Also, at the end of ch. 28, Thorne indicates that Cooper entered the tesseract at a point right along the outflying singularity (the fact that he and TARS passed through the outflying singularity was necessary to the plot since this allowed them to gather the "quantum data" about the singularity--the other answer of mine I linked to above discusses this as well). Quoting from ch. 28:

As for your second question about why Cooper wasn't still moving at high speed inside the tesseract, this isn't explicitly addressed by Thorne, but perhaps it's designed so that the part of it that intersects our 3D space can match velocities with any desired object in that space. This is suggested by the fact that the tesseract was semi-permanently docked to Murph's room, even though the room was on the surface of a spinning and orbiting planet, and also that Cooper was able to interact with Amelia Brand in a later scene, giving her a "handshake". So we could imagine that inside the black hole, the tesseract's intersection with our 3D space was moving along a course that not only would lead it to meet Cooper right at the outflying singularity, but also would lead to its velocity being approximately matched to Cooper's at that moment (although not perfectly matched, since Thorne says in the above quote that Cooper continued falling for a while within the tesseract before he 'comes to rest'). Alternately, since the bulk beings were supposed to have mastered the control of gravity (again see this answer of mine for details), perhaps they used that to adjust his speed once he entered the tesseract.

I'll address question #1 first. I think this one has to do with the fact that Cooper and TARS actually reached one of the singularities inside the black hole, where TARS was able to gather the needed "quantum data" (my answer here covers why this was necessary), and that was the point where they were rescued from the black hole by the "tesseract" which had been created by the beings living in the higher spatial dimension. This bit of dialogue between Cooper and Romilly on Mann's planet is key:

The "tesseract", meanwhile, is supposed to be a piece of technology created by the beings (possibly descended from humans) who live in the extra spatial dimension, the "bulk". This idea of an extra extended spatial dimension is based on a real physics theory, the Randall-Sundrum model--see my discussion in this answer for more details. The tesseract is shaped like a four-dimensional hypercube (that's what the word 'tesseract' means, in fact), so each of its "faces" is a 3D cube, just like each face of a 3D cube is a 2D square. In ch. 29 Thorne describes how the tesseract can "dock" one of its faces to our ordinary 3D space, which in the Randall-Sundrum theory is a 3D brane sitting in the 4D space of the bulk (for anyone familiar with the classic "math fiction" story Flatland, I think this docking of a higher-dimensional structure with our space is meant to be analogous to how the 3D sphere was able to materialize in the 2D universe by having one of its cross-sections in the 2D plane). Also, at the end of ch. 28, Thorne indicates that Cooper entered the tesseract at a point right along the outflying singularity (the fact that he and TARS passed through the outflying singularity was necessary to the plot since this allowed them to gather the "quantum data" about the singularity--the other answer of mine I linked to above discusses this as well). Quoting from ch. 28:

As for your second question about why Cooper wasn't still moving at high speed inside the tesseract, this isn't explicitly addressed by Thorne, but perhaps it's designed so that the part of it that intersects our 3D space can match velocities with any desired object in that space. This is suggested by the fact that the tesseract was semi-permanently docked to Murph's room, even though the room was on the surface of a spinning and orbiting planet, and also that Cooper was able to interact with Amelia Brand in a later scene, giving her a "handshake". So we could imagine that inside the black hole, the tesseract's intersection with our 3D space was moving along a course that not only would lead it to meet Cooper right at the outflying singularity, but also would lead to its velocity being approximately matched to Cooper's at that moment (although not perfectly matched, since Thorne says in the above quote that Cooper continued falling for a while within the tesseract before he 'comes to rest'). Alternately, since the bulk beings were supposed to have mastered the control of gravity (again see this answer of mine for details), perhaps they used that to adjust his speed once he entered the tesseract.

And for your third question, two observers who fall into a black hole can indeed continue to exchange signals back and forth as they fall. If you just want some confirmation this is true, see this page by the physicist Andrew Hamilton about the experience of someone falling through the event horizon, which says "Persons who appear to us to be inside the Schwarzschild bubble have passed inside the horizon of the black hole. If they are sufficiently close to us, then we can communicate with them, but they must be close, for there's not much time left before we hit the central singularity, not much time left for light signals to travel between us." But if you want some understanding of why this is the case, I gave my own conceptual explanation for this in this answer on the physics stack exchange--as I said there, I think the issue is easiest to understand if you use a "conformal" spacetime diagram like a Penrose diagram or a Kruskal-Szekeres diagram. In such diagrams, time is shown on the vertical axis and the radial space dimension is shown on the horizontal, and anything moving at the speed of light will be represented as a straight line inclined 45 degrees from the verticle. By the same token, the world line of any object moving slower than light (i.e. the line or curve showing its position as a function of time) will always have a slope that's closer to vertical than 45 degrees. Then the key to understanding why you can't escape the event horizon once you've crossed it is that it is also represented as a straight line at 45 degrees from the vertical--so in effect, in the coordinate system the diagram is based on, the event horizon is moving outwards at the speed of light, so once you're inside it there's no way to overtake it and cross back out unless you could move faster than light yourself. But there's no problem changing direction and moving back in the "outward" direction, or sending a light signal in the outward direction to communicate with a friend who's falling alongside you. This answer to another question by John Rennie includes a Kruskal-Szekeres diagram of a falling object sending light signals in both the inward and outward direction while inside the horizon of a non-rotating black hole:

And for your third question, two observers who fall into a black hole can indeed continue to exchange signals back and forth as they fall. If you just want some confirmation this is true, see this page by the physicist Andrew Hamilton about the experience of someone falling through the event horizon, which says "Persons who appear to us to be inside the Schwarzschild bubble have passed inside the horizon of the black hole. If they are sufficiently close to us, then we can communicate with them, but they must be close, for there's not much time left before we hit the central singularity, not much time left for light signals to travel between us." But if you want some understanding of why this is the case, I gave my own conceptual explanation for this in this answer on the physics stack exchange--as I said there, I think the issue is easiest to understand if you use a "conformal" spacetime diagram like a Penrose diagram or a Kruskal-Szekeres diagram. In such diagrams, time is shown on the vertical axis and the radial space dimension is shown on the horizontal, and anything moving at the speed of light will be represented as a straight line inclined 45 degrees from the verticle. By the same token, the world line of any object moving slower than light (i.e. the line or curve showing its position as a function of time) will always have a slope that's closer to vertical than 45 degrees. Then the key to understanding why you can't escape the event horizon once you've crossed it is that it is also represented as a straight line at 45 degrees from the vertical--so in effect, in the coordinate system the diagram is based on, the event horizon is moving outwards at the speed of light, so once you're inside it there's no way to overtake it and cross back out unless you could move faster than light yourself. But there's no problem changing direction and moving back in the "outward" direction, or sending a light signal in the outward direction to communicate with a friend who's falling alongside you. This answer to another question by John Rennie includes a Kruskal-Szekeres diagram of a falling object sending light signals in both the inward and outward direction while inside the horizon of a non-rotating black hole:

I'll address question #1 first. As explained in Ch. 26 of The Science of Interstellar by physicist Kip Thorne (who was the movie's science consultant), a realistic version of a rotating black hole like the one in Interstellar would actually have more than one singularity. There is of course the one at the center, which Thorne says would likely be a type of singularity known as a BKL singularity. This type of singularity would rip apart all objects with ever-more-violent oscillations in the tidal forces, which are gravitational forces that act differently on different parts of an extended object (an astronaut's feet being pulled more strongly than his head, for example) and therefore have the effect of stretching and squeezing it. The wiki article on the BKL singularity only talks about it in the context of the Big Bang, but you can look at this article for a discussion of how BKL singularities would apply to black holes.

Now, even if an extra bulk dimension does exist in reality as postulated by the Randall-Sundrum theory, I don't know if it would actually be physically possible for anything to leave our 3D space from a point inside a black hole's event horizon and escape the black hole entirely. Theories involving a "bulk" dimension say that the gravitational force can travel from objects in our 3D space into the bulk (whereas other forces like the electromagnetic force are supposed to be confined to our 3D space), so anything in the bulk should still be affected by gravity and perhaps this would mean that the event horizon would extend up into the bulk, so that jumping into the bulk from the black hole's interior wouldn't actually allow something to escape the black hole's event horizon (though I found this paper saying physicists have had difficulties describing how black holes would work in the Randall-Sundrum theory, so it may be something of an open question). But since Thorne said specifically that the tesseract was in contact with the singularity itself when it picked up Cooper, this would mean quantum gravity could be involved in its un-docking from our 3D space, so without a theory of quantum gravity we can't really say for sure whether escape from a black hole would be possible at that point. If you want to imagine a way to escape a black hole in the context of a speculative science fiction story that's just trying not to explicitly violate any known physics principles, this seems like a reasonable way to go.

I'll address question #1 first. I think this one has to do with the fact that Cooper and TARS actually reached one of the singularities inside the black hole, where TARS was able to gather the needed "quantum data" (my answer here covers why this was necessary), and that was the point where they were rescued from the black hole by the "tesseract" which had been created by the beings living in the higher spatial dimension. This bit of dialogue between Cooper and Romilly on Mann's planet is key:

ROMILLY: I have a suggestion for your return journey.

COOPER: What?

ROMILLY: Have one last crack at the black hole. Gargantua's an older, spinning black hole - what we call a gentle singularity.

COOPER: Gentle?

ROMILLY: They're hardly gentle, but their tidal gravity is quick enough that something crossing the horizon fast might survive...a probe, say.

COOPER: What happens to it after it crosses?

ROMILLY: Beyond the horizon is a complete mystery - who's to say there isn't some way the probe can glimpse the singularity and relay the quantum data? If he's equipped to transmit every form of energy that can pulse - X-ray, visible light, radio -

TARS: Just when did this probe become a 'he'?

ROMILLY: TARS is the obvious candidate. I've already told him what to look for.

As explained in Ch. 26 of The Science of Interstellar by physicist Kip Thorne (who was the movie's science consultant), a realistic version of a rotating black hole like the one in Interstellar would actually have more than one singularity. There is of course the one at the center, which Thorne says would likely be a type of singularity known as a BKL singularity. This type of singularity would rip apart all objects with ever-more-violent oscillations in the tidal forces, which are gravitational forces that act differently on different parts of an extended object (an astronaut's feet being pulled more strongly than his head, for example) and therefore have the effect of stretching and squeezing it. The wiki article on the BKL singularity only talks about it in the context of the Big Bang, but you can look at this article for a discussion of how BKL singularities would apply to black holes.

Now, even if an extra bulk dimension does exist in reality as postulated by the Randall-Sundrum theory, I don't know if it would actually be physically possible for anything to leave our 3D space from a point inside a black hole's event horizon and escape the black hole entirely. Theories involving a "bulk" dimension say that the gravitational force can travel from objects in our 3D space into the bulk (whereas other forces like the electromagnetic force are supposed to be confined to our 3D space), so anything in the bulk should still be affected by gravity and perhaps this would mean that the event horizon would extend up into the bulk, so that jumping into the bulk from the black hole's interior wouldn't actually allow something to escape the black hole's event horizon (though I found this paper saying physicists have had difficulties describing how black holes would work in the Randall-Sundrum theory, so it may be something of an open question). But since Thorne said specifically that the tesseract was in contact with the singularity itself when it picked up Cooper, this would mean quantum gravity could be involved in its un-docking from our 3D space, so without a theory of quantum gravity we can't really say for sure whether escape from a black hole would be possible at that point. If you want to imagine a way to escape a black hole in the context of a speculative science fiction story that's just trying not to explicitly violate any known physics principles, appealing to the mysteries of quantum gravity seems like a reasonable way to go.