I'll address question #1 first. As explained in Ch. 26 of *The Science of Interstellar* by Kip Thorne, a realistic version of a [rotating black hole][1] 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][2] which would be rip apart all objects with ever-more-violent oscillations in the [tidal forces][3], 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][4] for a discussion of how BKL singularities would apply to black holes. 

In addition to the central singularity, theoretical analysis suggest there'd be two others which Thorne labels the "infalling" and "outflying" singularities. These occur when, due to the particular way that [time dilation][5] occurs inside the black hole, waves and matter which fell into the black hole at many different times will converge into a thin sheet, with the tidal forces approaching infinity as you cross the sheet. However, unlike the central singularity, these two singularities 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 shape of an object by a finite 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][6], is expected to become inaccurate in regions of spacetime with sufficiently high energy density, known as the [Planck scale][7], and in these regions it's thought that a theory of [quantum gravity][8] would be needed to make accurate predictions ([string theory][9] 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][10]--there wasn't much I could understand, but if you look at the little primer [here][11] 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')

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][12]--see my discussion in [this answer][13] for more details. The tesseract is shaped like a four-dimensional [hypercube][14] (the word 'tesseract' is just another name for a four-dimensional hypercube), 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 it can "dock" one of its faces to our ordinary 3D space (for anyone familiar with the classic "math fiction" story [*Flatland*][15], perhaps it's analogous to how the 3D sphere was able to materialize in the 2D universe by having one of its cross-sections in the 2D plane0. Also, at the end of ch. 28, Thorne indicates that Cooper entered the tesseract 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 I linked to above discusses this as well). Quoting from the book:

> 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. 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--these theories say that gravitons emitted in our 3D space can travel into the bulk, so anything in the bulk should still be affected by gravity (though I found [this paper][16] 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 docked on the singularity itself, this would mean quantum gravity would be needed; 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 matched to Cooper's at that moment. Alternately, since the bulk beings were supposed to have mastered the control of gravity (again see [this answer][17] 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][18] by the physicist [Andrew Hamilton][19] 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][20] 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][21] or a [Kruskal-Szekeres diagram][22]. 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][23] 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][24] 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:

![enter image description here][25]

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 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.


  [1]: http://en.wikipedia.org/wiki/Rotating_black_hole
  [2]: http://en.wikipedia.org/wiki/BKL_singularity
  [3]: http://en.wikipedia.org/wiki/Tidal_force
  [4]: http://www.einstein-online.info/spotlights/singularities_bkl
  [5]: http://en.wikipedia.org/wiki/Time_dilation
  [6]: http://en.wikipedia.org/wiki/General_relativity
  [7]: http://en.wikipedia.org/wiki/Planck_scale
  [8]: http://en.wikipedia.org/wiki/Quantum_gravity
  [9]: http://en.wikipedia.org/wiki/String_theory
  [10]: http://arxiv.org/abs/1109.5139
  [11]: http://jila.colorado.edu/~ajsh/insidebh/penrose.html
  [12]: http://en.wikipedia.org/wiki/Randall%E2%80%93Sundrum_model
  [13]: http://scifi.stackexchange.com/a/73096/22250
  [14]: http://en.wikipedia.org/wiki/Hypercube
  [15]: http://www.eldritchpress.org/eaa/FL.HTM
  [16]: http://arxiv.org/abs/1409.0817
  [17]: http://scifi.stackexchange.com/a/73096/22250
  [18]: http://casa.colorado.edu/~ajsh/singularity.html
  [19]: http://casa.colorado.edu/~ajsh/home.html
  [20]: http://physics.stackexchange.com/a/158148/59406
  [21]: http://en.wikipedia.org/wiki/Penrose_diagram
  [22]: http://en.wikipedia.org/wiki/Kruskal%E2%80%93Szekeres_coordinates
  [23]: http://en.wikipedia.org/wiki/World_line
  [24]: http://physics.stackexchange.com/a/111925/59406
  [25]: https://i.sstatic.net/lIikg.gif