***I was hoping somebody could clear up these cosmological enigmas for me.***
Don't get your hopes up. I don't think anyone will clear these issues up for you.
***I am not sure how to visualize these small dimensions that are supposed to exist. I am infering from the article that space is 3 dimensional, time is the 4th dimension, and we are "encased" in a 5th dimension. The other dimesions are smaller than the diameter of a proton. What does that mean? What are these dimensions a part of? Are we existing in them or are they just all around us? I don't know how to picture it!!***
I have strong opinions on all these matters. I disagree with almost everything I have read by other people as to the answers to your questions. I have found nobody who will debate with me on my disagreement with prevalent scientific opinion on these questions.
Therefore, if I answer your questions, you cannot claim to know anything as a result. If you are looking for the accepted scientific answers, mine will only confuse you.
But...since you have given me the opening, I will once again try to explain my view of this subject. You may take it for what it is worth (exactly what you paid for it) and maybe you can find some flaws in my logic. Here's what I think.
1. There is evidence for, and most scientists will admit, the fact that our 3-D space is "bent" or "curved". The common explanation is that space is curved in the presence of mass.
2. There are several ways of "bending" or "curving" space that are not equivalent. To illustrate three of these, imagine both a flat sheet of paper and a flat sheet of rubber, like part of a balloon. a), you may bend either of these sheets in the fashion of origami. That is, it may be bent without stretching it. You can do this with either the paper or the rubber. b), you can stretch part of it, but keep the whole thing in the original plane. You can do that with rubber but not with paper. c), you can bend it and stretch it, as in stretching the rubber sheet over part of a ball. You can't do that with paper either.
3. There are even more complex ways of "bending" that you can't do with either rubber or paper some of which involve the space passing through itself, as in a Klein Bottle.
4. In mathematics, there is a concept called a 'metric' which is defined on a space. A metric is a function of two variables, each of which is a point in your space, and the value of the function is taken as the definition of the distance between the two points.
5. If you can define a metric on a space such that the Pythagorean theorem holds everywhere on that space using the metric as the definition for distance, then the space is said to be Euclidean, and the metric is said to be a Euclidean metric.
6. It is not always possible to define a Euclidean metric on a space. For example, 2.c) above does not admit a Euclidean metric in the two dimensional sheet.
7. In that case, however, it is possible to define a Euclidean metric on the space, but only by moving up one dimension. I.e. the ball may be considered to be embedded in a three dimensional space, and the Euclidean metric is defined on the three dimensional space. Be careful, though, because now distances between two points on the ball are not measured on the surface of the ball along a great circle, but they are measured by a "straight" line that goes through the interior of the ball intersecting it at the two points.
8. In common discussions of this subject, people are not always clear about what they mean by 'flat'. Sometimes they mean 2.a). Other times, they mean 2.a) or b). Other times they mean the space admits a Euclidean metric. You have to be careful but authors typically are not careful in this respect.
9. In 2.b), the paper is not flat, in the sense that it does not lie flat on a table, but it still admits a Euclidean metric. All triangles on it still contain 180 degrees (a necessary and sufficient condition for the Pythagorean theorem to hold.)
10. In mathematics there is the concept of a manifold. A manifold is a space which is embedded in a space of at least one higher dimension. For example, the surface of a sheet of paper is a 2-D manifold embedded in the 3-D space of the room. The edge of the sheet is a 1-D manifold embedded in the 3-D space of the room. The edge is also a 1-D manifold embedded in the 2-D space of the sheet.
11. I thought I learned about a theorem in topology class or differential geometry class that says you can't have a bent space unless that space is a manifold embedded in a higher dimensional space.
12. I have been unable to find a reference to that "theorem" and I have been unable to find anyone who will confirm its existence.
13. I remain convinced that there is such a theorem, not only because of my vivid recollection of that professor telling us about it, but also because if you think about it intuitively, it makes sense.
14. So,...I conclude, if our 3-D space is bent (in the presence of gravity or for any other reason) it must be a 3-D manifold embedded in at least a 4-D space.
15. As seems intuitively obvious, the nature of a flat (i.e. "unbent") space including the possible structures and features that it might contain, is independent of whether or not it is an embedded manifold.
16. If a space is bent in such a way that it does not admit a Euclidean metric, then I am convinced (see 13.) that it must necessarily be an embedded manifold.
17. It is possible, via measurements made strictly within a space, to determine whether or not it is Euclidean. (e.g. by finding a triangle whose angles do not sum to 180 degrees).
18. We have found our 3-D space to be non-Euclidean.
19. From the foregoing, we can conclude that if our 3-D space is an embedded manifold, other than manifestations of its curvature, all other features of the space should be independent of the fact that it exists embedded in a higher dimensional space.
20. Therefore, the existence of the extra dimensions would be undetectable from within the manifold.
21. One example is that inverse square laws, which you would expect to hold in a 3-D space, would also be expected to hold in a 3-D manifold embedded in a 4-D or even a 40-D space.
22. Dimensions do not have "sizes". I.e. it makes no sense to talk of a "large" dimension or a "small" dimension.
23. A manifold may be seen as a structure in the space in which it is embedded. (Familiar examples are intake and exhaust manifolds, jars, pipes, etc.)
24. It makes sense to talk about the "size" of a manifold if it is seen as such a structure.
25. Every science writer I have read who has commented on this subject from Stephen Hawking to Brian Greene, talks as if we cannot explain why we can't "see" or access higher dimensions if they really exist. To explain why not, they suppose those extra dimensions are "rolled" or "curled" up so tightly that they become too "small" to detect.
26. It is my firm opinion that you would not expect to be able to detect features or structures that exist in higher dimensional space which embeds your space as a manifold because all structures involved in detecting (eyes, telescopes, cameras, objects which reflect light, etc.) are 3-D structures in the manifold.
27. I see absolutely no reason to suppose extra dimensions are "rolled" up, but I see extreme unnecessary complexity introduced as a result of doing so.
Now, hopping down off my soapbox, let me give you my answers to your questions, Scott.
***These models are in some way based on the "M-theory" which is part of superstring theory.***
That is not a question, but I should point out that superstring theory is part of M-theory, not the other way around.
***I am not sure how to visualize these small dimensions that are supposed to exist.***
You are supposed to visualize them as higher dimensional analogs of rolling up a 2-D sheet of paper into such a thin tube that it appears to be a 1-D line -- at least if you stand back far enough when you look at it. (I think this is nonsense. There is no analog to "standing back far enough" without introducing even more, even "bigger", dimensions in order to get that vantage point, and "rolling it up" makes no difference in how that sheet appears to any "Flatlanders" who might live on it.)
***I am infering from the article that space is 3 dimensional, time is the 4th dimension, and we are "encased" in a 5th dimension.***
Yes, that's right, where "encased" means "embedded".
***The other dimesions are smaller than the diameter of a proton. What does that mean?***
That's what they say. I say that is nonsense and is meaningless. I say that dimensions don't have sizes, and "rolling" one up doesn't make it "smaller".
***What are these dimensions a part of?***
That is a good question about all dimensions whether or not they are accessible and familiar, or inaccessible or even "rolled up". Nobody knows what they are part of. I suspect they are simply part of the thoughts of ... (pick one: God, G.O.D., GOG, the mindful universe, all that is, the one-and-only mind in the sky, etc.)
***Are we existing in them or are they just all around us?***
What do you mean by "we", white man? (No offense intended. I just like that old joke.) Our universe, including our bodies, is simply a 3-D manifold embedded in a space of more than 3 dimensions, and we don't have much, if any, access to anything outside of our manifold. (Purported PSI phenomena may be an exception to this.)
***I don't know how to picture it!***
Picturing it is tough. How are you at seeing the 3-D images in those random dot pictures? I think that with enough mental exercise along these lines, we can begin to picture some of the features of 4-D and even 5-D space. I made a number of wire and wood models of 4-D (and even one 5-D) hypercubes, and by gazing on those models and imagining the features you know must be there by geometric inference, you can picture it to some extent.
***Also, according to the article, inflation also plays a part in all of these colliding branes. According to one of the scientists, inflation addresses the problem of the universe being uniform and flat. I thought the universe was curved, and I was under the impression that mass was not distributed evenly throughout the universe.***
This is a good example of the mixing up of terms like "inflation", "colliding", "branes", "uniform", "flat", "curved", etc. that, in my opinion are nothing but confusing nonsense without being specific about what those terms mean.