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.
In addition to the central singularity, theoretical analysis suggest that for rotating black holes there'd be two others, which Thorne labels the "infalling" and "outflying" singularities. These occur when, due to the particular way that time dilation occurs inside the black hole, waves and matter which fell into the black hole at many different times will converge at the "inner horizon" at the same moment--this is a second horizon found in rotating black holes below the outer event horizon, which is not predicted to be present in non-rotating, uncharged black holes (the inner horizon is also often referred to as the Cauchy horizon, and is also theoretically predicted to be present in non-rotating but charged Reissner-Nordstrom black holes). From what I have read elsewhere, the fact that there is a type of singularity here is related to the fact that electromagnetic and gravitational waves become infinitely blue-shifted at the horizon, meaning their wavelength approaches zero as they approach it--this is mentioned for example in the section "Penrose diagram illustrating the cause of mass inflation" at the bottom of this page.
However, unlike the central singularity, these two singularities at the inner horizon may be "gentle" ones in the sense that not only can individual particles theoretically pass through them and continue on the other side, but the period when the tidal forces become really large might be brief enough that they'd only distort the relative positions of the atoms in an object by a finite amount, an amount which might not be enough to rip apart a solid object like a human. Ultimately though, physicists can't really say what would happen when crossing these singularities even from a theoretical perspective, because the theory they use to analyze tidal forces and other spacetime distortions, Einstein's theory of general relativity, is expected to become inaccurate in regions of spacetime with energy densities that reach a scale known as the Planck scale, and in these regions it's thought that a theory of quantum gravity would be needed to make accurate predictions (string theory is an attempt to create such a theory, but it's incomplete).
Here's Kip Thorne discussing both the infalling and outflying singularities in chapter 26:
If you fall into a spinning black hole such as Gargantua, lots of other stuff inevitably will fall in after you: gas, dust, light, gravitational waves, and so forth. That stuff may take millions or billions of years to enter the hole as seen by me, watching from outside. But as seen by you, now inside the hole, it may take only a few seconds or less, due to the extreme slowing of your time compared with mine. As a result, as seen by you this stuff all piles up in a thin sheet, falling inward toward you at the speed of light, or nearly the speed of light, or nearly the speed of light. This sheet generates intense tidal forces that distort space and will distort you, if the sheet hits you.
The tidal forces grow to become infinite. The result is an "infalling singularity" ... governed by the laws of quantum gravity. However, the tidal forces grow so swiftly (Poisson and Israel deduced) that, if they hit you, they will have deformed you by only a finite amount at the moment you reach the singularity. ... Because your body has been stretched and squeezed by only a finite net amount, when you reach the singularity, it is conceivable you migh survive. (Conceivable but unlikely, I think.) In this sense, the infalling singularity is far more "gentle" then the BKL singularity. If you do survive, what happens next is known only to the laws of quantum gravity.
In the 1990s and 2000s, we physicists thought this was the whole story: A BKL singularity, created when the black hole is born. And an infalling singularity that grows afterward. That's all.
Then in late 2012, while Christopher Nolan was negotiating to rewrite and direct Interstellar, a third singularity was discovered by Donald Marolf (University of California at Santa Barbara) and Amos Ori (The Technion, in Haifa, Israel). It was discovered, of course, via an in-depth study of Einstein’s relativistic laws and not via astronomical observations.
In retrospect, this singularity should have been obvious. It is an outflying singularity that grows as the black hole ages, just like the infalling singularity grows. It is produced by stuff (gas, dust, light, gravitational waves, etc.) that fell into the black hole before you fell in ... A tiny fraction of that stuff is scattered back upward toward you, scattered by the hole's warpage of space and time, much like sunlight scattered off a curved, smooth ocean wave, which brings us an image of the wave.
The upscattered stuff gets compressed, by the black hole’s extreme slowing of time, into a thin layer rather like a sonic book (a "shock front"). The stuff’s gravity produces tidal forces that grown infinitely strong and thence become an outflying singularity. But as for the infalling singularity, so also for this outflying one, the tidal forces are gentle; They grow so quickly, so suddenly, that, if you encounter one, your net distortion is finite, not infinite, at the moment you hit the singularity.
(If anyone knows enough physics to follow any of the details, the 2012 paper discussing the outflying singularity is here--there wasn't much I could understand, but if you look at the little primer here on Penrose diagrams for different types of black holes, you can then compare with the authors' Penrose diagram for a realistic rotating black hole in Fig. 4 on p. 17 of the paper, which shows the infalling singularity as a red dotted line, and the outflying singularity as a solid red line labeled 'Shockwave'. Also note that the "Discussion" section on p. 18 talks about comparing "two different types of null singularity that develop at the inner horizon", confirming that this is where these two singularities are located.)
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:
In my science interpretation, as the Ranger nears the outflying singularity, it encounters mounting tidal forces. Cooper ejects just in the nick of time. Tidal forces tear the Ranger apart. Visually, it splits in two.
At the singularity's edge the tesseract awaits Cooper—placed there, presumably, by bulk beings
Apparently the tesseract is then able to un-dock and leave that region of our 3D space, travel through the bulk, and later dock itself to Murph's bedroom in the past. In ch. 29, Thorne says:
In my interpretation of the movie, the tesseract ascends from the singularity into the bulk. Being an object with the same number of space dimensions as the bulk (four), it happily inhabits the bulk. And it transports three-dimensional Cooper, lodged in its three-dimensional face, through the bulk.
Now, recall that the distance from Gargantua to Earth is about 10 billion light-years as measured in our brane (in our universe, with its three spatial dimensions). However, as measured in the bulk, that distance is only about 1 AU (the distance from the Sun to the Earth) ... So, traveling with whatever propulsion system the bulk beings provided, the tesseract, in my interpretation, can quickly carry Cooper across our universe, via the bulk, to Earth.
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To match what is shown in the movie, I imagine this trip is very quick, just a few minutes, while Cooper is still dazed and falling. As he comes to rest, floating in the large chamber,the tesseract docks besides Murph's bedroom.
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The back face of the tesseract coincides with Murph's bedroom. I'll explain that more carefully. The back face is a three-dimensional cross section of the tesseract that resides in Murph's bedroom in the same sense as the circular cross section of a sphere resides in a two-dimensional brane
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.
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:
As noted, the dotted line is the event horizon and the blue line is the world line of an observer falling in, and the two pink lines are light signals sent in both directions. Both the observer and the two light signals will inevitably hit the singularity which is represented by the red curve, but prior to that, if you imagine a second blue curve next to the first representing a second observer falling alongside the first, there would be no problem with them exchanging a few more pink light rays back and forth before hitting the singularity.