1. 걸다 is used when you hang something on a hanger. So you'll hang your clothes on a hanger. 옷을 옷걸이에 걸다. 달다 is used when you attach something on like a wall. So you'll hang a painting on the wall. 그림을 벽에 달다. So 걸다 is more temporary and 달다 is more permanent.
2. 깨끗하다 is used when it's clean and not messy. e.g., 집이 깨끗히다. House is clean and not messy. 청결 is clean as in hygienic or sanitary.
3. 따뜻하다 is when an object is warm or whether is warm. e.g., 호빵이 따뜻하다 or 날씨가 따뜻하다. 따스하다 is a bit sentimental. e.g., 따스한 봄바람. warm spring breeze but more in a sentimental way. Or 따스한 어린시절의 추억 fond/warm memories from childhood.
4. 갑자기 is suddenly. 노래가 갑자기 끝나다. 문득 is also suddenly/abruptly but it's pretty exclusive for thoughts. Thoughts that come up to your mind suddenly/abruptly.
When I have that kind of questions for English, I like to just type the word or expression in the google search box. Cuz then I can see how other people have used the word/expression in different contexts in their websites/blogs. So that way I can kinda pick up the connotation or nuance. A lot of times I end up at urbandictionary.com too hahaha There's no Korean version of urbandictionary but you can use 네이버/다음 or google to see how Koreans actually use those words. Hope it helps!