Checked exceptions are basically monads but with all the disadvantages of monads but none of the advantages. There is a reason no language since has copied them.
No they don't, not even when they're monads (you need monad transformers to compose monads since they don't universally compose).
swap :: Either e2 (Either e1 a) -> Either e1 (Either e2 a)
swap (Left z) = (Right (Left z))
swap (Right (Left y)) = (Left y)
swap (Right (Right x)) = (Right (Right x))
join :: f1 (f2 (f1 (f2 a))) -> f1 (f2 a)
-- join = fmap join . join . fmap swap -- if you have swap
swap is generally trivial (see above) for any sane error-reporting monad, although might be a bit more difficult if you're shoving error-handling logic into them.
* Use return values and let people deal with the boilerplate ala Optional in java. We use Optionals to replace null values and IMO the boilerplate is 100% worth it. I've also used return values that can encode possible errors in Java.
* Use unchecked exceptions and expect people to understand the methods they call and the exceptions those methods can throw, which should be documented in javadoc instead of a throws clause. This needs to happen regardless, as methods can throws unchecked exceptions the caller might need to be aware of, so this in reality doesn't involve extra work.
For me, either of these solutions is preferrable to checked exceptions.