"What led you to become a teacher?" -- 한국말로 뭐예요?
I'd like to say both "Why did you decide to [do X]?" and "What led you to [do X]?"

The two questions are a little bit different:
"Why did you decide to (become a lawyer)?" = Why do you like your job? What are your reasons for choosing it?

But the person may not like his job or have any reasons, so it's common to ask instead:
"What led you to (become a housesitter)?" = What happened before you got your current job?

The person may say:
"Well, I used to be a lawyer, but I lost my job. After that I decided to become an artist, but I couldn't make any money as an artist. So I got a job as a waiter in a restaurant, and after a year the owner promoted me to manager. But I didn't like being a manager, and I didn't have any free time for my artwork, so I told my boss. Since he trusted me a lot, he offered me a job housesitting for him two weeks every month while he is out of town for business. However, I never imagined I would have this job."

Are these questions common in Korean culture?

Question

"What led you to become a teacher?" -- 한국말로 뭐예요?
I'd like to say both "Why did you decide to [do X]?" and "What led you to [do X]?"

The two questions are a little bit different:
"Why did you decide to (become a lawyer)?" = Why do you like your job? What are your reasons for choosing it?

But the person may not like his job or have any reasons, so it's common to ask instead:
"What led you to (become a housesitter)?" = What happened before you got your current job?

The person may say: 
"Well, I used to be a lawyer, but I lost my job. After that I decided to become an artist, but I couldn't make any money as an artist. So I got a job as a waiter in a restaurant, and after a year the owner promoted me to manager. But I didn't like being a manager, and I didn't have any free time for my artwork, so I told my boss. Since he trusted me a lot, he offered me a job housesitting for him two weeks every month while he is out of town for business. However, I never imagined I would have this job."

Are these questions common in Korean culture?

Answer

Yes. I think both are common question/answer, i can totally understand about this situation.
 
-Why did you decide to (become a lawyer)? : 왜 변호사가 되기로 결정했어요? Or 왜 변호사가 되고 싶었어요? (Why do you want to be a lawyer?)

-What led you to (become a housesitter)?: 어쩌다 or 어쩌다가 housesitter가 되었어요? 
I think '어쩌다or 어쩌다가' is signifies little negation.

You can use '왜' as both positive or negative way:  왜 변호사가 되었어요?  왜 housesitter가 되었어요? (i think we can't say housesitter is not good but just use it in order to help you to understand more easier and i guess that's why you give like this example story :-)

But i think we do not use that much '어쩌다/ 어쩌다가' as positive way.    
For example:
-How did you hurt it? : 어쩌다 다쳤어요? 
-Did you quit your job? What happend? : 어쩌다가 일을 그만두셨어요?/ 일을 그만두셨어요? 어쩌다가!

Hope it helps you!

Answer

I'm not a native, but I think I can give an answer that's really close to what you want. You could say like:

어떻게 해서 선생님이 되기로 하게 되셨어요? or
어떻게 해서 선생님이 되게 되셨어요?

The context that I've heard "어떻게 해서" used in is like "How did you end up joining this club?", and the person would say like "Well I was really interested in ___, so I was searching on the internet and I ran across an advertisement and...."

I'm sure that a native speaker will come and say which of these is better.

Answer

"Why did you decide to [do X]?"
왜 [XXX]가/이 되고(or 하고) 싶었어요?

예) 왜 선생님이 되고 싶었어요?
     왜 한국에 오고 싶었어요?

"What led you to [do X]?"
어떻게 하다가 [XXX]가/이 되었어요?

예) 어떻게 하다가 선생님이 되었어요?
     어떻게 하다가 한국에 오게 되었어요?

Answer

What led you to become a teacher? = 왜 선생님이 되기로 했어요? or 무엇때문에 선생님이 되기로 결심했어요?

Normally "왜" directly translates into "why" in english, but I think, in this case, it is used more as the "what" in "what led you to become a teacher".