-거든 is basically an informative ending. 나 지금 나가봐야 되거든. 내 대신 일 좀 봐줄 수 있어? I've gotta go out now. Could you cover for me? - you're informing the listener that you must go out. 내 대신 일 좀 봐줄 수 있어? 나 지금 나가봐야 되거든. Could you cover for me? Because I've gotta go out. (This sounds a bit more polite.) This ending can carry a sarcastic or cheeky or slightly angry tone depending on the context. The most typical one is "됐거든", means "No, thanks!"(sarcastic). A : 좀 도와줄까? Do you want help? (smug expression on A's face) B : 됐거든! No, thanks! A : Are you Chinese or Japanese? B : 나 한국 사람이거든!

Maybe.. the whole paragraph is like this. A : 너 오늘 시간 좀 있어? B : 아니 없어. 오늘 소개팅 있거든 = 아니 없어. 오늘 소개팅 있기때문에 (안되) 거든 is the same as "because" , in this example. 자기 밥값은 자기가 낸다구 As explained above, this might be used to sounds cute. But I think.. just understand this sentence "자기 밥값은 자기가 낸다." And.. Sorry I cannot explain "구" because it is used to underscore some feeling but it varies in the respect of situations. These days, some teenagers say like this "자기 밥값은 자기가 낸다규~~" or "자기 밥값은 자기가 내는거삼" In these examples, "다규~~", or "거삼" sounds funny and perhaps maybe.. cute.

거든 usually means "because" but nowadays people use it instead of -어 to be 'cute' ^_^ 다구 is an informal form of 다고. In your example, probably the full sentence would be "자기 밥값은 자기가 낸다고 (better yet 내기로) 하자." It is like 'that' in "that subject + Verb" which is used to make a noun phrase. However, again nowadays this expression is used to suggest something in a cute way.