#5, on the other, is at the opposite end of the wrong spectrum, i.e. it is harmless. I just chalk it up to naivete and a lack of having read the scriptures much. When I’m teaching (though I’m not a gospel doctrine teacher) I don’t usually address such issues head on, if they come up. Sometimes I’ll make a side comment like “different people have different views on that one” and move on. I don’t see my job as teacher to expose the “shocking truths” of the gospel. Of course it shouldn’t be shocking that Jesus drank alcoholic wine or, as it seems to me from D&C 27:5, that the saints will drink alcoholic wine with Jesus at the second coming, but I don’t feel that Gospel Doctrine class or Elder’s Quorum is the place to set out my case for such beliefs. Further I worry about the risk of offending one of the saints. And I think that some of the discussion on various blogs related to Mormon teachings (this blog being no exception) show just how excited people can get over a discussion about such topics. I actually think such discussions should take place somewhere, but not during our Sunday Church Meetings. So the risk/reward payoff in correcting such errors doesn’t make it seem to me worth addressing typically.

“Why is there such diversity of belief among Mormons? (Note that most items on my list are Big Deals.) Is this is desirable?”
Lay ministry. I like Eric’s comment about the scriptures in #2.

“How do you, as a teacher, identify what is “core doctrine” so you teach only that, seeing that we disagree on doctrines with profound implications?”
I try to stick to concepts that are centered on Christ and scripturally based. As little as I like the teacher’s manual, I think it can assist in identifying “core doctrine”. Interestingly in D&C manual I couldn’t find any discussion of polygamy. Big issue, but the church seems to want to table the issue for now, at least in Gospel Doctrine. I will comply.

“How do you respond to other Mormons when they voice a belief that you reject—when you are the teacher? when you are the student? when you are in a casual (i.e., non Church) setting?”
Teacher? I try to anticipate some of the more common thoughts that I don’t agree with and just ask additional questions that may help people consider other ways to look at the topic. I may skip the questions and just present a secondary way to look at things, but I have not found it necessary to say, “No, that is wrong, this is right.”
Student? Depends on how open the classroom discussion is. In my CES training class our teacher brought up a doctrine concerning Kolob I had never heard of. He presented it in a way that nobody would really want to comment on it. (“Most of the church doesn’t know this, but I know all of you already know that Kolob…) I was really bothered by my teacher for that. First thing I did was go home and start search for a reason he might believe that, then I asked people who I trusted a little more, and then I talked with classmates to share what I had learned.
Casual setting? I prefer that people think through their position, so I usually ask questions, do you really mean this? or this?

“How do you teach these “non-unanimous topics” to your children? Do you teach them the truth-as-you-know-it-period or do you acknowledge the diversity of thought? Is this different than how you teach in the classroom? Why?”
I don’t think I have talked much about some of the non-unanimous topics with my kids. They are still young. I would like to think that I’ll share different things that they may here and share what my stance is, particularly if it’s in the realm of being agnostic about it.

Check out D&C 130:22

She also claims that the lost 10 tribes are riding around in UFOs. She prayed about it, so it must be true.

Couldn’t it just be that God chose to appear in the form of a man for some reason, perhaps so Joseph would find the experience more comprehensible? God appeared to Moses as a burning bush, but that doesn’t mean God physically has leaves or branches or is on fire.

We only have the narrative theology of the scriptures, and not a systematic theology. How to piece the narrative together is no easy task. And yes, this is very desireable. It is the only way to be if you are to base your beliefs on what is accepted revelation than on man-made creeds.

How do you, as a teacher, identify what is “core doctrine” so you teach only that, seeing that we disagree on doctrines with profound implications?

I try and stick to the scriptures and official sources (tttf, lds.org) when I am in a teaching capacity.

How do you respond to other Mormons when they voice a belief that you reject—when you are the teacher? when you are the student? when you are in a casual (i.e., non Church) setting?

I might ask them where this belief is supported in scripture. I might go as far as suggesting a resonable opposing view that is about as well supported. I might point out why I think the belief is ‘flawed’.

How do you teach these “non-unanimous topics” to your children? Do you teach them the truth-as-you-know-it-period or do you acknowledge the diversity of thought? Is this different than how you teach in the classroom? Why?

I usually teach them what I believe and why. I may tell them that I don’t really know all the details.