Hokay, starting in no particular order:
My workflow style tends to be pretty linear because I have a high reliance on spatial and temporal locality--I'm most aware of things I've worked on recently, or that are similar to things I've worked on recently, and those things tend to get more work until they're done. This is odd because my learning style is *not* linear--I spread out and learn about bunches of different aspects at once (seventeen Wikipedia tabs open at a time, etc.) and synthesize those bits, but when it comes to doing things... Things tend to get done either immediately or not at all. Which is partly why I usually write back to you fairly quickly. If a week goes by without a reply from me you can pretty much count on it having fallen off my queue, and I may not write again for months unless you remind me.
Similarly, I tend to forget people exist if I'm not interacting with them regularly. There are a small handful of people who stay in my low-level awareness pretty much permanently--old friends, cousins, "kindred spirits" if you will--but I've so often had the experience of being recognized by old acquaintances whom I didn't myself recognize that I've just resigned myself to the embarrassment. Although that's partly also my bad facial-recognition capabilities, especially outside of context; I don't even always recognize my own sisters (true story), and my cousin M.J. once caught me on our college campus to say hi, and it took me over sixty seconds of guessing to figure out who she was. Pretty awful.
Come to think of it, I guess even the small handful of kindred spirits only stay in my awareness because I've got a low-level background task refreshing my memory of them. For instance, I've just today renewed my deal with my cousin B. to write her one letter every month. This is the third year in a row I've been doing this, and one of my motivations is that B. is one of my favorite people in the world (smart, pretty, lots of integrity, assertive and a bit fierce to her peers but kind to those who are weaker than her, which is how I think people should be) and I don't want to forget about her now that I've moved to Seattle and don't see her frequently. (Related: are you familiar with the notion of a "monkey-sphere"? Primates have trouble keeping track of too many relationships; humans average higher than other primates but still no more than one hundred fifty or so relationships per person. Google "monkeysphere".)
Yes, I probably am a ponderer, although I'm at least as much of a talker--I do my best thinking when there's another person in the loop, whether it's a class I'm teaching or a friend I'm planning on sharing my thoughts with. (And sometimes I get halfway through a line of thought and go, "Actually, I'm not really buying this argument, are you? I think it's wrong.") Recently I've been thinking about goals, and pondering who it is I want to be in 2050 and what kind of goals are worth making. I won't share those thoughts, though, since (brutally frank here) you're not really inside my monkeysphere, or at least not the inner circle of people that I trust enough to discuss personal feelings and thoughts with. Sorry, the exclusion is expedient. I'm more than happy to discuss with you anything that I would discuss with an artificial intelligence, though.
Islam is interesting to me. Actually, Islam tends to be interesting to a lot of Mormons because we have a surprising number of beliefs in common--probably not any more than we have in common with Jews or Catholics or Eastern Orthodox Christians or ancient Egyptians, but not necessarily less either. I was talking to my co-worker E. fairly recently (she's Muslim), and she happened to mention her belief that we lived as spirits before being born on Earth. I was like, "Huh? How do you know about that?" The scriptures say that the Lord raises up wise men unto all nations, speaking in their own language, and sometimes I wonder how much he has said to whom. Of course it's also possible that the belief in question was simply passed down from the early Christians (Coptic Egyptians or somebody), which got transmitted along with a sort of residual respect for the Bible. Or maybe Muhammad really did speak with an angel--it's hard to know without having read the Qu'ran, which I haven't. (Maybe some day I'll get around to it.) The one thing I do know is that whatever else he was, Muhammad was not called as a prophet because prophets always, always testify of Christ, in some language or another, and Muhammad didn't. But that still doesn't explain why Muslims know so many interesting truths.
You want to know the plot of the play, hmmm? Tell you what, since you're asking me, I'll give you _my_ understanding of the scriptures, references available upon request but with the understanding that it's always possible I'm misunderstanding something--the general thrust of this is quite orthodox, but my way of looking at things is sometimes idiosyncratic. (In other words, other Mormons probably wouldn't disagree with me but they'd probably also never tell it this way on their own, and especially not starting where I'm starting.)
There are two kinds of things which exist or can, things which act and things which are acted upon. The scriptures generally call the first kind "spirit," or sometimes "intelligence," and where the second is referred to at all it is called "element" or "matter." It's not completely clear whether intelligence is innumerate or intrinsically individualistic, but it's what you and I are, and it can neither be created nor destroyed. Intelligence has always existed and always will, although it can learn and progress and grow. Our story has no beginning, but we'll begin at the first point the scriptures discuss, which is when God, who is also an intelligence, called his children together to announce that he had a plan whereby they would receive bodies made of element, learn from experience about good and evil, and have the opportunity to progress to become like he is.
22 Now the Lord had shown unto me, Abraham, the intelligences that were organized before the world was; and among all these there were many of the noble and great ones;
23 And God saw these souls that they were good, and he stood in the midst of them, and he said: These I will make my rulers; for he stood among those that were spirits, and he saw that they were good; and he said unto me: Abraham, thou art one of them; thou wast chosen before thou wast born.
24 And there stood one among them that was like unto God, and he said unto those who were with him: We will go down, for there is space there, and we will take of these materials, and we will make an earth whereon these may dwell;
25 And we will prove them herewith, to see if they will do all things whatsoever the Lord their God shall command them;
26 And they who keep their first estate shall be added upon; and they who keep not their first estate shall not have glory in the same kingdom with those who keep their first estate; and they who keep their second estate shall have glory added upon their heads for ever and ever.
27 And the Lord said: Whom shall I send? And one answered like unto the Son of Man: Here am I, send me. And another answered and said: Here am I, send me. And the Lord said: I will send the first.
28 And the second was angry, and kept not his first estate; and, at that day, many followed after him.
The one "like unto God" was Jesus Christ, the firstborn and greatest of the children of God. ("Firstborn" is probably an approximation of the actual concept since, as noted before, intelligences have always existed and so there is no being "born" in the conventional sense, but it's a pretty good approximation.) The second, who rebelled, was Satan, or rather the entity who became known as Satan (in Hebrew, "the accuser"). We know from other scriptures that the controversy in heaven was that Jesus said there would be certain souls who would not be saved, and Satan said he could save them all, and coveted God's glory in order to force that to occur; and when Jesus was chosen to be the Only Begotten and the Savior of mankind, Satan rebelled and was cast out. (By the way, Satan was wrong. It is impossible to force intelligences into perfection; you cannot subjugate them to your will and simultaneously bring them to perfection. The very idea is nonsensical--empowering them without giving them power to rebel against you.)
And that, really, is the plot of the play. We're here to taste the bitter in order to know the sweet, and to learn to forsake evil by doing good. There are some complications. One is that none of us, except Jesus, was or is strong enough to always do the right thing once we were placed in our current circumstances; this is both the reason why Adam and Eve were warned not to partake of the fruit of the tree of knowledge of good and evil (Moses 3:17, "But of the tree of the knowledge of good and evil, thou shalt not eat of it, nevertheless, thou mayest choose for thyself, for it is given unto thee; but, remember that I forbid it, for in the day thou eatest thereof thou shalt surely die.") and the reason why the Lord, knowing that we must partake of that knowledge in order to progress, sent Jesus to suffer alongside us, each and every one of us, vicariously and in all our pains and trials and sins, so that having "descended below all things" he would also have the power and the right to forgive our sins to one another and to bring those who hear and obey his voice back to the fold.
Not incidentally, he also has the power to correct all ills which are merely that of element--resurrection is the reuniting of body and soul in a perfect restoration, never to be divided again and never again to suffer pain or injury. Everybody who ever lived will be resurrected eventually--one of the reasons Jesus showed himself to his apostles after his resurrection was to show them how literal the promise was, not of being some nebulous spirit but the actual person restored again. Ills of the spirit are more problematic, but everyone who desires to do righteousness will receive as much capacity for righteousness as he desires to receive, while those who desire to do wickedness must remain as they choose to be:
32 And they who remain shall also be quickened; nevertheless, they shall return again to their own place, to enjoy that which they are willing to receive, because they were not willing to enjoy that which they might have received.
But the point of life on Earth is to be tested and to learn, and even though it's hard sometimes we know that even the worst of problems are not really as bad as they seem, any more than a disastrous experience on the first day of kindergarten is really a disaster to the grown-up person you are now. Revelation 21:4 says, "And God shall wipe away all tears from their eyes; and there shall be no more death, neither sorrow, nor crying, neither shall there be any more pain: for the former things are passed away."
There, that's the basic plot. We know some other things--various things that must happen before the time is right for Christ to return to Earth in glory, and some of the things which will happen after he comes and what life will be like here then, and a little tiny bit about the far future--but now you know the most important parts of the play. Except that you really don't know them and can't know them until the Lord confirms it to your heart directly--you're never responsible to believe in mere words--but at least you know what I believe and know. Any questions?
Wow, I'd better go to bed soon. I see I've overlooked a couple of your questions so I'll answer them briefly: yes, the people going door-to-door may be missionaries for the Church of Jesus Christ of Latter-day Saints. Or they may be shoe salesmen or Jehovah's Witnesses or something. It depends upon which people you're seeing. :-) If they're missionaries they'll have little black name tags with their names and the name of the church on it. Whatever I know about talking about my beliefs I actually didn't learn on my mission, really, partly because the Filipino culture is different (so what I learned there isn't applicable here) and partly because I wasn't old enough. I think I just learned it by being alive for long enough and by living my life (as opposed to anyone else's, which would have taught different skills).
"Are Mormons required to go on a mission?" you also ask. I don't really know how to answer that question--the whole point of life is that no one is ever required to do anything, strictly speaking. Serving a mission is also a privilege and you're required to meet certain standards. At the same time, it's a really, really good idea, especially for the young men (who, as I've mentioned before, will serve as priests and church leaders throughout their lives), and often a good idea for young women. Thus, the answer to your question is probably that, for the young men at least, it's expected and encouraged to prepare for a mission to the same degree that most people are encouraged to go to college--I have male friends who didn't serve serve missions, but they're the exception--and for young women it is one possible option which they are encouraged to exercise if they feel they should. (And we are grateful for the sister missionaries. They have unique gifts and can do some things the elders can't. All the ones I served with were really neat.) (By the way, "elder" is a priesthood title and doesn't necessarily have much to do with age--all male missionaries are ordained elders and are addressed as "Elder So-and-so" while on active duty, so to speak.)
There, I think that covers all your questions. If I've overlooked any, you know what to do.
 Leo Rosten quote: "I learned that it is the weak who are cruel, and that gentleness is to be expected only from the strong."
"The presentation or 'gift' of the Holy Ghost simply confers upon a man the right to receive at any time, when he is worthy of it and desires it, the power and light of truth of the Holy Ghost, although he may often be left to his own spirit and judgment." --Joseph F. Smith (manual, p. 69)
"Inasmuch as ye have done it unto one of the least of these my brethren, ye have done it unto me."