The use of preposition with time

Question

Going off of this question:

"This semester" or "In this semester"?

Having read through the question, its answers, and the discussion in comments, I would like to get to the bottom of this grammar issue because that discussion is obviously inconclusive. Lambie's answer is helpful in pointing out which is more natural in one particular situation, but doesn't provide a thorough explanation. The other answer gives a speculation/conclusion without explaining, and I share Ronald Sole's query into the reasons behind that speculation/conclusion. That discussion leaves me (if not the asker of that question and other learners) still perplexed.

I understand and agree with Lambie's answer that "This semester, he has finally abandoned his habit of reading comic books in class" sounds more natural than "In this semester, he has finally abandoned his habit of reading comic books in class." However, I wonder if that intuitive principle applies universally in all other situations. And I am curious about the WHY whether the answer is yes or no.

Lambie has helpfully pointed to the distinction between time and preposition phrase. What grammatical functions do these two have in a sentence? Is "in this semester" adverbial in the above sentence? What about "this semester"? I thought they both were adverbial phrases. Am I correct?

Does in this semester also have the same quality in these sentences:

What did you learn in this semester?

Later in this semester, we are going to learn calculus. (Examples are from Google searches)

Can these sentences be reworded without the "in"?

What about these sentences? Do they sound natural? Would a different version, either with "in" tagged onto the sans-preposition versions or taken off from the preposition versions, sound better or stilted? And why is that?

In this semester we want to busy ourselves with the basic problems of a general phenomenology of consciousness. (The Basic Problems of Phenomenology by Edmund Husserl, translated by Ingo Farin and James G. Hart)

(In) August last year, I attended a summer conference in California.

(In) The following year, I took a trip to a musical festival in New York City where I met a bunch of like-minded people.

Answer 1

I can't tell you about time and prepositional phrase, but I can tell you about your examples:

What did you learn in this semester?

Later in this semester, we are going to learn calculus.

You can remove the "in". However, if you were to replace "this" with "the" (e.g. Later in the semester), you would need the "in".

In this semester we want to busy ourselves with the basic problems of a general phenomenology of consciousness.

The "in" here makes the phrase sound more formal and correct, which is good in a textbook. Without the "in", the author would be emphasizing this semester (not last semester or the next semester), which is probably not the author's intent.

(In) August last year, I attended a summer conference in California.

You definitely need the "in" here.

(In) The following year, I took a trip to a musical festival in New York City where I met a bunch of like-minded people.

With or without "in" makes no difference.

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Edit: Because I was curious, I made a whole table. Stack Exchange doesn't support tables, so you can check it out on pastebin here.