Convolution output signal is very large, do I need to scale inputs or outputs?
In practice you would make sure that your filters don't have very large gains, at least not over a large frequency range. The filter you're using in the given example has a maximum gain of almost $26$ dB (at DC), and the gain is no less than $12$ dB for any frequency. I.e., you're applying a flat gain of $12$ dB, and on top of that you boost a relatively wide band around DC by another $14$ dB. This is not what a typical filter for audio signals would do, unless you make sure that the input signal is small enough and the additional gain is really needed.