There is nothing wrong with those phrases, and I'm sure people use them, but they're a bit "literary" for most people's spoken English. They would probably say, "tell a secret" rather than "disclose a secret." It's more likely that these phrases would be written rather than spoken. On a more general note, asking if people say things is really impossible to answer: people say everything, even incorrect things, so the answer would always be "yes."

Yes, you can use them

"Clear up a mystery" sounds a little weird to me, but something like "I need to clear up this mystery" sounds fine, so yes, it should be usable. Usually, though we'd just say "solve a mystery". "Disclose a secret" is perfectly fine.