1. What makes a poem a poem is the ability to make the reader feel something. I know that can happen with prose, as well, but it seems to me it must happen with a poem. As has already been mentioned, a poem is different in form from prose--the normal rules of writing just don't apply. That doesn't mean a poem can't have form or punctuation, but it doesn't have to. One of the books I teach from is called Sound and Sense, which comes from the idea that poetry should match its sound with its meaning (sense).
2. Rhyme, form, sound.
3. Yes, without a musical rhythm poetry is dead. Poetry is a sense of expression, completely based on imagination which we portray in words and rhythm is used by adorning words. ... Music (words + melody) can be seen as poetry.
4. Yes they could!