I wouldn't use it for a person, I would say 'getting into' The new bookmark I bought is so cute, and I put it in a book of selected poems — I think I might be getting into poetry! Hope this helps :)

It's not so clear, so better not use it like that. Instead, you could say "I may soon fall in love with poetry" which means roughly the same thing, but is a bit more explicit, for clarity.

Yes, you can use it that way so long as you are aware of the underlying meaning of "fall for". Look carefully at the two definitions given by your dictionary. It is important to realize that (1) and (2) are not two different meanings. They are the same. Meaning (1) says that the person has worked some magic on you that made you ("tricked you to") fall in love with them. It is the romantic version of being tricked! So, it is appropriate to say you "fell" for a book of poems if you mean to say that the book tricked you (in a good way, of course) to adore it. It would be an especially appropriate thing to say if you did not at first like the book at all but you slowly realized (as it worked its magic on you) that it was great and it won you over. The language of magic finds its way into many expressions about love. For example, we say "she cast a spell on me", "she bewitched me", "she is charming" (or "she is a charmer"). I would consider "I fell for her" to be a part of this family.

Your sentence makes little sense to me. How could a bookmark influence your attraction to poetry? A poet or a poem could do that, but even then I can’t imagine using ‘fall for’ to describe developing an appreciation for this form of literature. The positive meaning of ‘fall for’ really applies to romantic love. Using it in this case is very imprecise.

You can check the Oxford or Merriam-Webster dictionary for reference.