In the USA, one is 'discharged' from a hospital, meaning the hospital no longer has charge of (responsibility for) them. Hope this helps :)

Yes, you can say it at a hospital. You can also say the opposite, like: "Because I had too much stress, I checked myself into the hospital."

In the UK we would not use "check in/out" with hospitals, only hotels.

In the UK we normally use discharged.

Yes, you can say that. But the phrase "check out" from a hospital is borrowed from "check out" of a hotel.