As far as I know, there is no common idiom like "Sticks and stones may break my bones..." in Russian. So you can say: "Слова меня не ранят" / "Словам меня не ранить" "Слова меня не сломят" / "Словам меня не сломить" Phrases with "словам" (2nd options here) sound more poetic. Or, if it's not for this song but for something more light and ironic, you can say "Колкость глаз не колет" (that's how Jack Sparrow's phrase - I guess it was just "sticks and stones..." - was translated in "Pirates of the Caribbean: The Curse of the Black Pearl")

I'm afraid that there are no any idioms with the same meaning if we are talking about "Words" and "the power of words". More to say, in the Russian language all idioms that I know have the opposite meaning :) We like to say that words are the most powerful thing against other people, so if you want to kill mentally a Russian person, just say to him something offensive :) But in the life speech you can hear the phrase "Ваши слова ничего не значат для меня" or "Ваши слова не имеют для меня никакого значения". It translates into English like "Words you said have no meaning for me". P.S.: Sorry, I asked my mother about it, and she said me the expression "Собака лает, ветер носит" or "Собака лает, а караван идет", which means "The dogs bark, but the caravan goes on".

Слова не могут расстроить/сломать меня.

Русской идиомы на этот случай припомнить не могу. В просторечии и в молодой среде скорее используют вариант "Словами меня не загрузить"... Вообще, эту мысль русские обычно выражают не прямо, а слегка иначе. Например: "Она мне много всякого наговорила. Но я не впечатлительный."