1.
The first question is about the tense we should use after "as soon as". In a sentence such as this:
I had left when the phone rang.
you need to use the past perfect in the second clause to show which action came first and which – second. However, when you use “as soon as”, the sequence is clear and it is normally a matter of preference which one to use, so both your examples will be correct. In American English the preference would normally be past simple. The past perfect would emphasize the fact that one action was complete before the other one occurred. (an explanation given in Grammar for Teachers by Andrea DeCapua)
2.
In the second pair of examples they are both correct again. It is unnecessary to use past perfect because the time is mentioned and the sequence of events is clear. Also, the actions are described in the order in which they occurred. You can use the past perfect if you want, to emphasize that one was before the other.
3.
The third question was about the sentence
He said that the moment he first met her, he felt something special and began to keep a diary.
The actual words the man said must have been:
"The moment I first met her, I felt something special and began to keep a diary."
When you report his words and begin with “He said”, the entire phrase shifts one tense back and becomes:
He said that the moment he had first met her, he had felt something special and had begun to keep a diary.
Although this is the grammatically correct sentence, it is very common that the past simple does not become past perfect in indirect speech. When reporting, native speakers tend to make present tenses past ("I am studying" - "She said she was studying") but very often do not care to make the past tenses perfect, as grammar books always teach us we should.
That is what makes both these sentences correct: "He said that the moment he first met her, he felt something special and began to keep a diary." and “He said that the moment he had first met her, he had felt something special and had begun to keep a diary.” (have a look at the end of this page)
Perfects are statives, not eventives, so there is no inherent ordering; quite the contrary, the “default” implicature∗ is that the events are simultaneous or overlap.
However, the implicature can be overridden (for instance, through adverbs like then); and by and large hearers/readers will understand the sentence to communicate what you call the “logical” order.
More to the point, perhaps, is the fact that the second perfect is not required, or even particularly desirable, once you have established the anteriority of the email. I would write it this way:
The meeting took place because he had sent her an email, which she then forwarded to other members.
∗ a useful technical term I've just discovered and have been looking for an excuse to share. It means something implied but not entailed. That is, it is a likely but not a necessary consequence of the context: you may infer it, but the inference can be ‘cancelled’
Best Answer
Yes: when the chronological order is clear we often use the simple past, as you surmised and @P.E.Dant confirmed. It may be informal but is probably more frequent.
Edit: To me, the fact that they don't accept "both" as an answer indicates an insistence on artificially treating language as a system with one tool for each job, as described below.
Because questions like this come up often, I'll express some general thoughts on language tests that may serve as useful pointers.
Tests are written to evaluate a supposedly ideal usage. By this one often means a usage that represents a close correspondence between the grammar and the meaning. We can logically deduce that both sentences must mean that the boy lost his ticket before he told you about it. But only the past perfect grammatically compels you to read that sequence. Writers of language tests want you to use linguistic knowledge as much as possible to work out an answer — not semantic knowledge, like "one cannot tell about X before X happens".
This is like the semi-mythical teacher with whom every English speaker has this exchange:
The word "can" works just fine here, but forcing the student to make a distinction between "can" and "may" in theory makes the question more precise. To some people, this qualifies it not as a correct choice but as the correct choice.
Tests are based on the implicit fiction that language is a perfect system designed to express meaning unambiguously, a system in which every tool has its "correct" use. But in real life, we rely on context and knowledge to close the gaps inherent in linguistic systems or produced by people. Test writers know that too, but for the purposes of evaluation they anticipate "ideal" uses of the tool — perhaps because such questions can be marked by computers.
Tests are written not to the sentence, but to the topic. To the untrained eye, it appears that you're being asked to find the ideal item to fill a blank in a sentence. But that's not how the question was written. The test writers knew the test had to cover the past perfect, so they created an ideal blank for that item. The catch is that the function is not always reciprocal. The kind of sentence the past perfect "prefers" may also be "preferred" by the simple past! Questions that have more than one correct answer result from the failure to notice such alternatives. A carefully written test will avoid such questions, but then the sentences will be artificial, because it's rarely the case that every word in a natural sentence is inevitable.
At the end of the day, you take tests with these things in mind. Tests can be useful for gauging your progress, but the more interesting gauge is actually that you correctly questioned the test. If you know more than the test, you should not try to learn from the test itself.*
* This is an interesting issue with tools like Duolingo: every question is both a test and a learning point. Therefore, you will learn a version of the language that primarily satisfies test requirements. It's hard to strike the right balance with such a system: Either the criteria will be too strict and your knowledge will be artificial, or they will be too loose and you won't be pushed to learn (or they will fluctuate between those extremes). Luckily, the brain is a linguistic marvel and will often notify you when something doesn't "smell right", so an attentive, savvy learner can still get quite a lot out of such systems.