"check out" is really only used casually and it means something like "come look at this really cool/interesting thing". I would say in all of the sentences you gave "check" works better but depending on the context "check out" might work as well in some. The only time I can think of where "check out" isn't used casually, is when you "check out a book from the library".

"check" = "detect errors in" "check out" = "look at"