The base form -게 되다 signifies coming to a certain state or ending up doing something, without saying the exact reason or course of events that led to it.
가게 됐어요 means the decision has been made for the person to go (either by themselves or by others). So it means they did not go yet, but they will in due time. -ㅆ어요 here describes a completed state, where ...
Think of those Korean verbs as having a optional preposition in them. They can be either transitive or intransitive. For example, "살다" translates to "to live (for);" when it is transitive, the object is the duration.
내가 법학을 삼 년을 공부했어!
...Wait, it works. ...Never mind. It's not an object. It seems that the "을" particle after a ...
"그가 중국에서 2년을 살았습니다."
In English, you can say either "He has lived in China for two years" (intransitive verb "live") or "He has spent two years in China" (transitive verb "spend"). 2년을 살다 is thus similar to "spend two years", both 살다 and "spend" (or "pass" or some other) ...
The first “였다” is “이었다"'s abbreviations.
Noun: noun which has “받침” -이었다(었다)
noun which doesn't have “받침” -였다
verb: behind verb which is “past tense” (except for ㅏ, ㅓ) -었다
verb which attach(use)”~하다”,”하다” -였다
되도록 by itself means "as much as possible", similar to 될 수 있는 대로, 될 수 있는 한, 가능한 한, 가급적이면, etc.
Your sentence can be interpreted in two ways.
[아무리 바빠도 되도록] 저녁 식사는 ... = We should have dinner with family so we can be as busy as possible (makes no sense).
아무리 바빠도 [되도록] 저녁 식사는 ... = No matter how busy we are, we should have dinner with family as much ...
I think it's misleading to ask when "아/어서" means "to" - they are both very versatile constructions that can be used in various situations. So, some situations can be nicely expressed with "아/어서", some others with "to", and some others with both.
In the examples here, "A to B" can be interpreted as "A in ...
Yes, rewriting them that way would be correct.
Aside from the meaning of "Because", "아/어 사" Can be used in the following ways:
As a clause connector in which the second action occurs after the first (similar to 고). This is usually added to 오다 and 가다:
우리는 집에 와서 바로 잤어요: We came home and went to sleep immediately
저는 당근을 칼로 잘라서 냉장고에 넣었어요: I ...