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 order to do B". "A-아서 B" can be interpreted as "A, and then (after that/as a result) B". Given these definitions, one may say that the aren't the same at all - however, as shown in the question, there are sentences where both can be used naturally.
E.g., "저는 학교에 가서 공부할 거예요." would mean, "I will go to school, and (as a result of going to school) I will study." But that's basically the same as "I am going to school to study." Nobody would ask "Wait, is studying a mere byproduct of your arriving at the school or is it the original intention which prompted you to go to school?" In most cases, that's a distinction without substance.
Similarly, "나는 친구들을 만나서 이야기했어요." means "I met my friends, and (as a result) I talked to them." which is the same as "I met my friends to talk." If you meet your friends, then of course you talk - nobody meets friends only to stare at each other. So, again, they mean the same.
The reason why "to" is translated to "아/어서" here is because it's more natural in Korean. If you say "나는 이야기하기 위해서 친구들을 만났어요." that sounds rather weird, as if you had a specific thing in mind that you must have told your friends.
Now let me examine your "transformed" sentences:
나는 한국에 와서 한국어를 공부할 거예요.
This sentence is wrong, because you're saying "I will come to Korea and study Korean," but if you're saying that then you aren't in Korea now, so you can't use "와서" - you have to use "가서".
Or you could say "나는 한국에 와서 한국어를 공부했어요" = I came to Korea and studied Korean.
Its meaning is a bit different from the original sentence, because the original sentence specified that the reason you came to Korea was to study Korean, but the new sentence made no such claim. It's possible that you came to Korea for a job, and then you later decided you wanted to study Korean.
나는 한국어를 공부해서 통역사가 될 거예요. = I will study Korean and become a translator.
This sentence is similar to the original but there's a subtle difference. The new sentence is more about your resolve: you decided that you will study Korean and become a translator. In particular, you can say this sentence even if you didn't even start learning Korean. (As of now, you just want to be a translator.) The original sentence, of course, clearly means you are already studying Korean.
나는 일찍 일어나서 운동해요.
Hmm I think this one is pretty much the same as the original. However, the original sentence is explicit that the reason you get up early is that you want to exercise. It's merely implicit here. For example, one can consider a contrived situation:
A: 저는 일찍 일어나서 운동해요. = I get up early and do exercise.
B: 정말 운동을 좋아하시는군요! = Wow, you must like exercises so much!
A: 아니오, 아침마다 옆집 개가 짖어서 깨는데, 깨서 할 게 운동밖에 없어요. = No, the neighbor's dog barks and wakes me up every morning, and there's nothing else for me to do once I wake up.