Is the complement of every non Turing recognizable language a Turing recognizable language?
Your proof is fine. There are also explicit languages which are neither r.e. nor co-r.e., for example the language of all total Turing machines (Turing machines halting on every input).

In fact, the language of all total Turing machines is $\Pi_2$-complete, which means, in a sense, that there is no better way to "solve" it than to run all on inputs and see whether the given machine halts on each of them (of course, you can't do this with a Turing machine, but perhaps with a more powerful device...). Take a look at the arithmetical hierarchy for more on this.