Both allora and poi can be translated as "then", but they are not synonyms.
Allora can have a cause-effect connotation, whereas poi is more temporal, meaning just "subsequently".
Quindi has a wider meaning and includes both of these.
Besides, allora also means "then" as in "at that time" (a quel tempo / a quei tempi).
It has the same meaning but 'allora' is a bit more general, 'at that moment' refers to a specific point in time and is a set phrase. For instance with 'he carried on coming home later everyday, then i began to understand', 'then' refers to a period of time rather than a single point
Which preposition has to be used depends on the first verb of the construction. They usually are A and DI, while PER is used in final sentences (don't know if that's what they are called in English, sentences expressing a goal/purpose).
Allora cominciai a capire = Then I started to understand.
Allora cercai di capire = Then I tried to understand.
Allora studiai per capire = Then I studied to understand (so that I could u.)
Mmmm but there is not a one to one relationship with words here. "So" has a meaning of "then" in this sentence, indicating a cause and effect. But then "so" can also mean "like that" just like "così", like just what happened without there being cause and effect. Using "so" here doesn't change the meaning of the sentence. But if the sentence was "Così ho cominciato a capire" I would probably translate as "And like that I began to understand".
Essentially, the Italian sentence mean: because of that event, due to what I heard, because of what somebody said, because of what I read, because of what I have (suddenly) been gone through, since I am in that determined situation or environment... And like that must refer to one of these cases (which it doesn't I think), and must not mean "For that reason". It 's not a simple, plain cause in fact, but it concerns something big, important, something you "touched with your bare hands". That is the subtlety what I was talking about, you see. It's not a simple cause-and-consequence situation, as the word "così" seems to suggest. If you want thing to fit perfectly, you got to analyse. The problem is that I may exxagerate then, I am aware. But strictly spoken, "così" is a little different connotation. You see what I mean, Hunter?
Thanks for updating your post and making it clearer, but it's still hard to understand, sorry. What I can say is adverbs can be tricky in that their meaning can often change based on the sentence that they are in, this might not be obvious until you try and translate from one language to another.
ALLORA ho cominciato/iniziato a capire. Allora takes the meaning of 1.From that moment, from then on. 2. Then. 3.At that moment 4. At that point. But staying near to the Italian sentence, Duo should choose only then/from then on. Because the other variants have a translation on their own. AT THAT MOMENT: Da quel momento ho cominciato a capire. AT THAT POINT: A questo punto, ho cominciato a capire.
for present perfect tense (passato prossimo), you will use either a form of "avere" (in this case "ho cominciato" for "i have started") or "essere" in combination with the verb form for things that happened in the recent past. "cominciato", as the passato prossimo conjugation for first person, isn't valid on its own.
I put "So I have begun to understand", which was marked wrong. "So I began to understand" was the 'correct' answer given. "Ho cominciato" surely translates as "have begun", and if not, why the Hell not? I do see there is a difference in meaning, but I don't see why it's not "have begun". I have not begun to understand!
In English "now i began to understand" has the same meaning as "then I started to understand" in that the speaker is reflecting back on a moment in time on an event. Duolingo doesn't understand this. But then/now I need to get an understanding of how "iniziare" and "cominciare" are used, as "initiate," "commence" and "begin" are often used in the same context in English.