Need to be able to encode a machine as syntax. This should be easy and obvious. Problems with no inputs are simply questions. All CFLs are decidable. Exist non-CF decidable languages.
All decidable languages are recognizable, by definition.
If L and not-L both recognizable, then both are decidable. By definition of machines, one will accept x. Since both are recognizable, that one will halt on x. Can decide using that info. Don't know questions' answers, but each decidable regardless. Can simply always answer yes or no, whichever is actually right.