I will explain as far as I know. (Although there might be little confusion due to my English skill (clearly not due to my Korean skill, since I am native Korean who read many books)).
As you know this means cold, but particularly for weather or space. Koreans usually say (in winter)
"날씨가 춥다."(Weather is cold) or they say "이 방은 춥다."(This room is cold)
but never say
"얼음이 춥다."(Ice is cold)
"춥다." usually implies negative meaning. If someone says "방이 춥다." then he/she means that its "too cold". (On the other hand, "차갑다, 차다" is quite neutral. They usually does not imply emotion.)
(2) 차갑다, 차다.
I think they are almost same. These word has general and objective meaning "cold". So you can say "얼음이 차갑다." or "얼음이 차다." to translate "Ice is cold." in Korean. I think "날씨가 차다." is acceptable but "날씨가 춥다." is more natural.
Also note that you can use this expressing feelings, like "그는 차갑다." , which means "He is cold-blooded."
*) You said "the last three words, which, I think, should be closer to the meaning "cool"", and I think it is plausible. The last 3 words are milder.
This implies positive meaning. Imagine in very hot summer you dive into cold water. Then you feel good with the coldness, and you can say "시원하다!".
Adverb form "쌀쌀맞게" can be used in many ways, but as adjective, I think this describes only weather or temperature. It is mild version of "춥다". Usually Koreans say "쌀쌀하다" when the weather is colder than they expected, but endurable.
Similar to (3) but bit more objective and calm. For example, if again you dive into cold water in hot summer you feel sudden cold, and say "시원하다!", but if you turn on your air-conditioner in your room and the temperature gradually decreases, you can say "서늘하다".
Also, 서늘하다 can be used in negative meaning, especially describing implied emotion. For instance, "나는 그의 서늘한 시선을 느꼈다" means "I recognized his cold eyesight", probably implying bit of negative meaning.
(6) 식다: This is nearly always used to express something that was once hot, but currently cold.
Ex: 이 국은 따뜻했지만, 식었다(This soup was hot, but now cold) or 나는 그를 사랑했지만, 이제는 마음이 식었어
Lastly, for other examples of the word "cold". There are "을씨년스럽다" or "음산하다", but their deeper meanings are bit difficult to explain. They involves some emotional coldness(e.g. for 귀신(ghost) or 흉가(haunted house)).