Can any information leave a black hole? Referring to this question, is there any information that can leave black holes? Are they causing a permanent information loss in our Universe?
Information cannot "leave" the black hole. There is no way (by our current framework of physics) that an entity inside the event horizon can send a signal out.

However, entropy can leave. The black hole has entropy proportional to its surface area, and this roughly corresponds to "the number of ways that black hole could have been created". The surface area decreases due to Hawking radiation, so it can release entropy back to the universe. Of course, the growth rate due to infalling mas is usually many orders of magnitude more than the shrinkage.