I think, and someone please correct me if I'm going wrong here, 공부하기로 could be "through studying". I think "기" goes with 공부하다 putting it in the "-ing" form. "로" is a ending particle meaning "way", "method" but flexible used so that if can be used in saying things like "to get to X take that street over there". But I'm not 100% sure about being right here. You'll have to look at the context of the text containing 공부하기로 to see if my take even makes sense.

펴다 = 1. open, unroll, spread out (a folded thing). 2. make a contention, logically/persuasively explain ones opinion, etc. (주장을 펴다, 논리를 펴다). 친구는 해괴한 논리를 펴기 시작했다 = My friend launched into a rather strange logic.

You could have posted four questions :-) Here' my answer for the first one. 신경쓰다/신경을 쓰다 = to mind/bother, pay attention mindfully. 신경쓰이다/신경이 쓰이다 = be bothered/annoyed by (쓰이다 is the passive of 쓰다). 쓰인단 말이야 = 쓰인다는 말이야 = I'm saying you're/it's getting on my nerve. -다는 말이야 = I'm saying(말이야) that ...(... 다는) = phrae of restating something (-다는 is a quote form which can be used for someone else's or one's own remark or thought.