adverbsinversionnegative-inversionpast-perfect
I found this sentence from my dictionary. I wonder why "had" is before "a moment".
Hardly had a moment passed before the door creaked open.
I also wonder about other words like never, once in a while, scarcely, etc. when they are at the beginning of a sentence.
The truth is that there are many situations and sentences which allow for either the past perfect tense or the past simple tense, with very little change to the sentence. The key difference between the past perfect and past simple tenses is the explicit order of two (or more) events in the past. There are of course many nuances in usage that make this a fairly complex topic, which I can only partly cover here. First, here is a brief overview of some of the relevant usages and rules of past simple and past perfect:
Past Simple
The third point is where the past simple tense overlaps with the past perfect tense, and is what I will be focusing on.
Simple tenses are often represented as a single dot on a timeline, such as the one below. In the case of past simple, this represents that the event is completed entirely in the past. If past simple is used to list multiple events, then we represent it with multiple unconnected dots in the past, arranged in chronological order.
e.g. I ate breakfast this morning and then my brother arrived.
Past Perfect
Perfect tenses are often represented by two dots with a connecting "bridge" on the timeline. This represents both the order (past perfect clause = left dot; past simple clause = right dot) and linking of the two events.
e.g. I had already eaten breakfast when my brother arrived.
Note: I find that timelines such as this often help my students understand tense usage better. This particular timeline is one I use often and is from this site.
The past perfect is not necessary if we are not 'going back' to the earlier past, and simply moving from one event to another.
This is correct. Past simple can be used to list multiple events in the past, but they should always be in order of occurrence, from longest ago to most recent (or have some other way of explicitly stating the order of events).
Yesterday I went to work, and then did some shopping.
If we use the past perfect tense instead, then we have more flexibility in the sentence structure while still being grammatically correct.
I had gone to work yesterday before I did some shopping.
or
I did some shopping yesterday after I had gone to work.
You can see that when using the past perfect tense we can list events in whichever order we prefer, rather than being restricted to chronological order.
If the sequence is clear, the past perfect is not needed.
Again, this is correct. However, there are nuances here. For the sequence to be "clear" you must either list events in chronological order as I mentioned before, or explicitly state the order or time in the sentence. This is often ignored by native speakers because the order of events can usually be inferred from the sentence without it being explicitly stated. In fact your example sentence shows this phenomena. I will again denote longest ago and most recent.
We showed the result of the survey we did.
We can in this case correctly infer that the survey was completed before the results were shown, it is simple logic. However, if you wish to be more clear (and grammatically correct) then it is better to write it in the past perfect tense.
We showed the result of the survey we had done.
I recommend my students to always use explicit ordering of events for the sake of clarity, so that the reader/listener does not need to use logic to deduce the order. Most prefer to do this by using the past perfect tense, or chronological order of events in the past simple tense.
The past perfect is optional [i.e. may be replaced by past simple] only when talking about an action at a specific time.
This "rule" is a combination of the usage rules for past simple and past perfect. As I mentioned before, one of the rules of past simple is that it must include a time, which can be either implied or specified. However, when the past simple is used to replace the past perfect, more information (i.e. a specific time) is needed to provide the same information to the reader/listener that the past perfect tense would have given.
Where you put an adverb or adverbial phrase depends on what it is you want to modify.
Before a sentence an adverbial is taken to modify the entire sentence; in other positions its scope may be more restricted.
For instance, in your first example, quietly may be placed at the beginning or after teacher if it modifies the main clause, but if it is placed at the end it will be understood to modify the subordinate clause to finish their game. The same constraints operate on your last example, except leasing ... cars loudly and convincingly makes no sense; consequently you will not be misunderstood if you put loudly and convincingly at the end. (But you should not do so, since this will create a momentary confusion.)
In the second example, which contains only a single clause, the adverbial may be placed at the beginning, after she, or at the end.
The third example is tricky. This sentence must be parsed You [can [speak {freely}], not You [[can speak] {freely}]: freely modifies only the lexical verb speak, not the permission or concession expressed by can, so freely must be placed after speak.
Best Answer
There are a lot of sentences (statements, not questions) that have inversion (the subject is placed after the first verb-form).
Negative adverbs as never/never in my life/seldom, rarely - but also adverbs as hardly, barely (used to indicate that two events follow immediately one after another) have inversion when these adverbs are placed at the beginning of the sentence.
Never in my life had I thought he would do such a thing.
Never have I heard such insolence.
Hardly had he turned round when a shot fell.
But there are a lot of other sentence types with inversion. You find this topic in a good grammar if you look for "inversion" in the register.