Turing machine that checks whether a given string is an output of a given machine and input Is there a Turing machine such that, given a description $\langle M \rangle$ of a Turing machine $M$, an input $x$ and a string $y$, computes whether or not $y$ is the output of $M$ input $x$?
My guess is that the answer is no because this might imply that the set of strings with Kolmogorov complexity greater than or equal to their length is decidable.
No, in fact any non-trivial semantic property of Turing machines is undecidable. This result is Rice's theorem.