Am I correct in assuming that there is a different word in Danish for each English definition of 'off' e.g. to mean spoiled, started or the opposite of 'on'?
It probably helps to think of it as "usage" not "definition." Both "spoiled" and "started" usages for "off" are slang. I find this perspective useful anyway, because if I can think of the original definition or usage of a word then it's easier to map it to a word in a different language. Many times there are words that came from the same root and knowing the foreign usage of the word of the same root helps me better understand the meaning of the English word. I'm a bit nerdy like that.
Yes, it's very common to use the definite article in this way about personal belongings and parts of the body
So would "They take their shoes off" be accepted?
Oh, wow. The audio for this sentence is not great, folks. The sound of skoene af is terrible!