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While knowing one will certainly ease learning the others it's rather misleading to say that Java, Haskell, Prolog, and Brainfuck all have the same structure. Their differences are significantly beyond syntax.

Yes!

I would go so far as to say that almost all the most important revelations I had programming-wise where when I learned a new language and suddenly gained a new way of looking at everything I know. And the nice thing about this is, you will keep these insights whether you stick with the actual language or not.

That said, there have been a few languages that have failed to do so. Almost invariably, I loathe working with them.

There are powerful insights to be gained by learning a functional language if your day job is coding in imperative languages.