I have found this article recently published in Classical and Quantum Gravity giving the exact solution of a scalar field in the Kerr-Newman metric. These authors also derived Hawking radiation for this black hole here. So, the question is do similar works exist for the Schwarzschild metric or, in this case, only approximate technique are used? I imagine that, after so much research on the Hawking radiation, there should be a large body of literature about this. Is it so?
You can find an explanation of scalar fields and associated quantum effects in the Schwarzschild background in chapter four of these lecture notes. The article also contains references which might be of use to you.