"건데 (것인데)" is used a lot in speech both in statements and questions.
It is however just one case of the more general ~ㄴ데 form (것이다 -> 것인데 -> 건데).
So you need to understand the ~ㄴ데 form.
~ㄴ데 primarily connects two parts giving a background (indirect, either causal or contrasting):
- 날씨도 좋은데 공원에나 가자: The weather being so good, why don't we go to a park.
- 비는 오는데 우산이 없다: It's raining and we don't have an umbrella.
It is also often used as a sentence ending, to make it smoother and more nuanced.
~는데 ending is similar to the plain endings (~아/어/야, ~다), but makes it sound like there's something left unsaid.
- 이거 너 주려고 만든 거야: This is what I made for you - sharp and clear cut.
- 이거 너 주려고 만든 건데: same, but sounds like there's more to it, whatever it might be.
- 어떻게 할 거야?: What are you going to do? (a specific question - sounds like it demands an answer)
- 어떻게 할 건데?: And? What do you think you'll do? (lighter, like a passing one in an ongoing conversation)
- 시험날은 다가오는데..: The test day is nearing.. (and? what am I doing?).
- 이래서는 안 되는데..: I shouldn't be[act] like this.. (but I can't help it).
* ~ㄹ까 is one of the common question forms, with two distinct usages:
1. an objective question(to oneself): 왜 얘가 안 올까? (Why is he not coming?), 여기가 어딜까? (What is this place?).
2. to suggest an action: 우리 이제 갈까? (Shall we go now?), 같이 식사할까? (Shall we dine together?).