My impression is "It's raining cats and dogs" is old-fashioned. Is that right? If I used it, would people think I'm 70 years old, or something like that?
It's not like saying "the cat's pajamas" or "23 skiddoo". We still use it — in informal speech. You're not likely to hear it on the TV weather report, though.
It's definitely an idiom: heavy rains have nothing to do with cats and dogs — who definitely do not fall from the sky. Here's a British website explaining the origin:
The first appearance of the currently used version is in Jonathan Swift’s A Complete Collection of Polite and Ingenious Conversation in 1738:
"I know Sir John will go, though he was sure it would rain cats and dogs".
While there may be generational differences, this ngram suggests that the phrase is not yet in serious decline, at least in the books Google samples. In fact, there appears to be a steady increase from 1975 through at least 2005.