Those terms, subject markers and topic markers, do not reflect the other use of 이/가 and 은/는. I would not use them.
Let's assume that you are talking to your close friend.
내 남동생이 예전에 노란 카나리아 한 마리를 키웠었는데, 아빠가 실수로 죽였어.
You are talking about the actions your younger brother and your father did on the yellow canary (You are not contrasting your brother with your father). The topic is the yellow canary, so you may rephrase this Korean sentence as
카나리아는 내 남동생이 한 마리 키웠는데 아빠가 실수로 죽였어.
(I have removed some redundant words). Nonetheless, you can use this rephrased sentence only either when your friend or you have just mentioned something about the canary or when both your friend and you have just heard what others say about the canary. This relates to the first and second mentions, which indicates the importance of the context in the Korean language.
Then, what is wrong with the following sentence?
내 남동생은 카나리아를 한 마리 키웠는데 아빠가 실수로 죽였어.
Now, your focus is more on your brother; the topic seems to be your brother. To your surprise, this sentence can be understood as
My brother had been raising a canary, but my father killed my brother by mistake.
Why is such interpretation possible? Words attached to 은/는 can be used not only as the subject but also as the object (as you can see in my past answers), and "아빠가 실수로 죽였어" itself does not have the object for 죽이다. Thus, 내 남동생 here is used as the subject for 키우다 and the object for 죽이다. This is how Korean works. Even if you clarify the object for 죽이다 as in
내 남동생은 카나리아를 한 마리 키웠는데 아빠가 그 카나리아를 실수로 죽였어.
it is somewhat confusing since your friend cannot be sure which one you want to talk about: your brother or the canary. By now, you should have noticed that the following sentence is usually incorrect:
내 남동생은 카나리아를 한 마리 키웠는데 아빠는 실수로 죽였어.
This sentence does not clarify the object for 죽이다. Your friend will have no idea what your father killed by mistake and probably ask what your father killed. Then, how about the following sentence?
내 남동생은 카나리아를 한 마리 키웠는데 아빠는 그 카나리아를 실수로 죽였어.
This sentence is okay to use if you would like to contrast your family members, but this is not what you mean, isn't it? The next sentence "왜냐면 아빠는 새장을 청소하려고 화학용 스프레이를 뿌렸거든" says the cause of the canary's death. For the two sentences to be connected to each other, you should use "내 남동생이 ... 아빠가 ...."
As for "왜냐면 아빠는 새장을 ...", "왜냐면 아빠가 새장을 ..." sounds a lot more natural as MujjinGun mentioned. It is because the topic has been the canary and this sentence relates to the canary's death.
If you have mentioned to your friend something about your father and would like to focus mainly on what your father did, you can use the following:
내 남동생이 카나리아를 한 마리 키웠는데 아빠는 그 카나리아를 실수로 죽였어.
왜냐면 아빠는 새장을 청소하려고 화학용 스프레이를 뿌렸거든.
In this case, the topic is your father.
A2: (I will skip this because MujjinGun already mentioned the correct reason.)
I guess you brought those examples from one of my answers, but I would say that the second translation is not precise. It should be "I don't understand at all what you mean." The third translation is technically incorrect because 태반 means "almost half," although a lot of people see it as "most." Anyway, I mentioned the requirement there:
Used right after a noun or adverb(ial)
I will explain a noun + 이/가 first.
이곳 사람들 태반이 글을 쓸 줄 모른다.
여기에 있는 사람은 대개가 어른이다.
As you see, this noun + 이/가 needs to be used as the subject or complement in this specific structure:
...(은/는) ...이/가 ...
where the noun attached to 이/가 refers to either part or all of the noun attaced to 은/는. With this structure, you can emphasize the portion. 태반 means "almost half" and 대개 means "most." The parentheses around 은/는 indicate that the omission of 은/는 could occur there. One more example is here:
언덕은 전부가 숲이었다.
which means that the whole hill was a forest. In another simple structure, you can use 은/는 instead to mean the same:
We could say that 은/는 is already an emphasizer, as it can mark the topic and also the comparison targets. What I meant with my categorization, however, was that the noun attached to 은/는 does not have to be the topic or the comparison targets. For example, the topic of the sentence
아무리 바빠도 밥은 먹자.
is "you" (your health, your work, etc.) rather than "meals" and this sentence is not about contrasting meals with other things. 은 is there in place of 을 to emphasize the "meals."
As for an adverb(ial) + 이/가 and an adverb(ial) + 은/는, I could not find a fixed rule, although the purpose is to emphasize the adverb(ial). I think you should remember what adverb(ial)s can have 이/가 and what adverb(ial)s can have 은/는.
저쪽에서 불빛이 번뜩이 비쳐 왔다. (Not 번뜩은)
네 말은 도대체가 무슨 말인지 모르겠다. (Not 도대체는)
손님이 올 것이니 밥을 넉넉히는 해라. (Not 넉넉히가)
가끔은 네가 생각난다. (Not 가끔이)
In these sentences, you can just remove the emphasizers without making them incomplete:
저쪽에서 불빛이 번뜩 비쳐 왔다.
네 말은 도대체 무슨 말인지 모르겠다.
손님이 올 것이니 밥을 넉넉히 해라.
가끔 네가 생각난다.