1. 가까이(adv and noun), 가까운(a), 가까운 곳 are more general than 근처 in its meaning of "close by". It comes from the adjectival verb 가깝다 (close), so it's never used as 가까 by itself. 근처 means "nearby place", and is mostly used for bigger structures, like houses, roads, neighborhoods, etc. For example, you can say 내 가까이로 와(come closer to me) or 나는 그와 가깝다 / 가까운 사이다 (I'm close to him), which cannot be said using 근처. 근처 is most natural when describing locations, like 이 근처에 편의점이 있나요? == 여기서 가까운 곳에 편의점이 있나요? (Is there a convenience store nearby?). 2. "Even you are too busy". Direct translation would be 너까지 너무 바쁘구나 / 너조차도 너무 바쁘구나 but I doubt these are actually said much. There are better ways to say it depending on what you mean. If there's no sarcasm involved, you'd say (난 몰랐는데) 너 참 바쁘구나 ((I didn't know) but you are quite busy). If it's sarcastic, 너도 바쁠 때가 있구나 ((I thought you never busy yourself with anything) but here you are, really busy) would work better. 3. 이제 and 지금 are similar in the broad sense of "now", but they are also different. 이제 often focuses more on the new situation that has come to be, while 지금 emphasizes "at this very moment" without any nuance. So 지금 끝났어 means "It finished just now" (some event/work just ended) and nothing else. 이제 끝났어 on the other hand can mean either 지금 끝났어 or "Now it's over" (there's no hope anymore), depending on the context and intonation.

1. Yes, their meanings are quite similar. 가까이 is a pure Korean word. It’s an adjverb meaning “near”. 근처 is a combination of two 한자 characters: 근 meaning “close/near” and 처 meaning “site”; in combination 근처 means “site close by”. “가까운 곳/가까운 데/가까운 장소” would be an exact equivalent of “근처” 3. Same thing. The two mean exactly the same; 지금 is originated from ancient Chinese and can be broken down to chinese characters, while 이제 is a purely Korean word. No noticeable difference in real usage apart from their sound value.