1. 나랑 상관 없어.
나 : a pronoun which refers to oneself.
-랑 : with
상관 : co-relation
없어 : there isn't
2. 관심 없어.
관심 interest
3. 말도 안 돼.
말 : word(s), language,
-도 : a postposition which mean "even"(here), "also"
안 : not
돼 : cannot be translated because it has lots of meanings in different situations. I think it means "work" here
4. 그렇겐 못 해.
그렇게 + 는 = 그렇겐
그렇게 : that way
는 : a postposition, it's used to emphasize here
5. 어쩐 일이야?
어쩐 : what
일 : work(noun), something
이 : used after the noun
야 : an ending, you can use it when you talk to your close friends(younger than you or your age). I use it while talking to some older friends of mine. Some are just 1~3 years older than me, some are really close to me although they are years older than me. It could be a case of myself.
6. 농담이겠지
농담 : joke
이 : a postpostion used after the noun, before the (pre)final ending.
겠 : a prefinal ending to mean "guess"
지 : a final ending, here it's used to imply the speaker thinks it must be a joke.
7. 그렇겠지
그렇- : an adjective verb stem, means be like that
Nah, you cannot use -이지 when you talking to random people. The reason is that -이지 implies you're talking to your close friends. But you can use -이지요 or -이죠, like 농담이죠 or 그렇겠죠. 요 is a respectful ending. ㅂ니다 is a formal ending. 요 sounds familiar, so use this ending when you want to get along with someone well. If you see "이", you can think it's used for nouns. Like 입니다, 이죠, 이네요, etc etc. For vebs, 합니다, 하죠, 하네요.
사랑하는 사람 the person I love
사랑한 사람 the person I loved