Yes, you can say 'The weather is crazy' as an informal comment. We'd understand it to mean extremely changeable and unpredictable. 'The weather being changing' doesn't make any sense.

Yes, you can. It's natural in informal U.S. English. Some usage examples from Google News: "Freezing rain and drizzle will fall in the Ozarks on Friday but will give way to rain and thunder on Saturday. Get the latest on our crazy weather by watching the forecast." During below-freezing temperatures a ice-cream store owner said "We've definitely had all of our regulars continue to come despite the crazy weather..." "Share your crazy weather photos with 5 News by emailing them to our social/digital reporter Phyllis Smith." A convenience-store manager: "Templeton told FOX23 they almost sold out of gloves, ice picks and beanies, as people try to predict this crazy weather."

Yes you can say "the weather is crazy". However, usually this means that the weather is currently being extreme (e.g. extremely hot, extreme snow storm, etc.) It does not usually signify that the weather is changing.