*-(은)ㄴ가/-는가 used for the question of/whether. Sometimes used like a quoted question/thought (to oneself): wondering to oneself, is it not a case of ... ?
-는가 is used for action verbs and -(은)가 is used for descriptive verbs for questions in questionnaires or exams.
한글은 언제 만들어졌는가? When was hangul invented?
속담과 격언은 어떻게 다른가? How are proverbs different from adages?
중력이란 무엇인가? What is gravity?
1) it's used when you ask someone about his or her past experience.
(usually uttered by higher or older person to lower and younger person
Was the movie interesting?
2) a rhetorical question, not really asking for an answer.
우리가 그동안 얼마나 지구를 더럽혀왔던가?
How (much) have we polluted the Earth?
3) when you talk to yourself, referring to what happened in the past.
내가 그런 말을 했던가?
Did I say that? (I don't remember or I shouldn't have said that)
-든가 links verbs together with ‘or’
Vst + 든가 … Vst + 든가 do either VERB1 or VERB2 (or the like). The -든가 here can be replaced by -든지.
아버지도 늦게 들어오시든가, 호텔에서 주무시든가 할걸요.
And (contrary to what you might be assuming) my father will probably either come home late or else sleep at a hotel.