세계 refers to all the countries on the Earth, whereas 세상 means all human societies that exist. You can use 세상 when saying something like "cruel world", but you cannot say 세계 here because what makes the world cruel is people, not the 세계 itself. 같이 is used in speech, but 함께 is used only in formal writing or literature. Yeah, 다 and 모두 both mean "all", 다 is more common in daily conversation. 처럼 and 같이 are "like" as in "It looked like a bat". 같은 is an adjective, therefore it's always used to describe a noun, for example, "천사 같은 아이 = a child that looks like an angel". 과/와 are used with nouns in formal writing. We don't use 그리고 as much as we do in English, instead we conjugate it with the verb, for example: 귀엽다 to be cute 저 고양이 귀엽다. the cat is cute. 저 고양이 귀엽고 사랑스럽다. the cat is cute and lovely. (이)랑 and 하고(both are used with nouns) are more common in conversation, they're what kids learn first. 또한 means a formal "also". About the difference between 또 and 다시... if someone says "또 해봐요" after you did something, they mean "Do what you just did again(one more time)". If they say "다시 해봐요", they mean "Do it again from the beginning", maybe because there might be something off. We say 다시 해봐요 when something didn't work the way it was supposed to.

1. 세상 - the world; (a) society , the public the age [era]; the days; the times 세계 - the world; the earth; the globe. a world; a society; circles 2. 같이 - (1) equally ex )모든 사람을 (똑)같이 대접해라 everybody. equally. treat. treat every body equally (2) at the same time 두 사람이 같이 출발했다. two guys at the same time depart(leave) two guys depart(leave) at the same time 2. its really hard to explain... 3. 같이 and 처럼 r almost same. so you can you any word . but there is one exception. but i think this can be used in poetry. so you dont have to know about that. 4. 와/과 can be used among nouns 사과와 바나나 apple and banana 그리고 is often used among sentences. 나는 미국인입니다. 그리고... im an american citizen. and...