Idioms are none of those, in my opinion; an idiom is just a phrase that has a meaning recognised in a speech community, but whose meaning is not obvious from the dictionary definitions of the individual words. They come from many sources- some cross from a specialised are into the wider comunity. 'Throw in the towel', meaning 'to surrender', comes from boxing, where a fight can be stopped by literally throwing a towel into the ring. Others come from individual's usage, particularly writers, so the 'milk of human kindness' comes, like many, from Shakespeare. There are many sources. The only are where they might come under one of those forms is the creation of phrasal verbs, which as generally regarded as idiomatic rather than idioms, but there you could say that syntax is at work.
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