For the "great kings of men" the most likely reason is the desire for increased power, potentially the ability to stave off old age and hold on to their domains.
For dwarves it would be the promise riches lost and riches to gain.
I don't think the elves took off the Rings - they were hidden from Sauron and those were the only three of the rings of power he didn't touch. However Frodo later sees Galadrial wearing her ring so they still wore them (Galadrial, Elrond & Gandalf (Cirdan)).
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"Dic, hospes, Spartae, nos te hic vidisse iacentes dum sanctis patriae legibus obsequimur."
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