DSP signal filtering and number formats
I suppose that conversions from signed 2's complement to Q15 and vice versa are necessary.

Yes &amp; No.  Q15 (signed) uses 2's complement, the bits are the same so no conversion is necessary. The difference between a Q15 and Q0 aren't the bit them selves but your interpretation of them. If you multiply two 16-bit numbers you get a 31-bit long result (one less than 32 because of a redundant sign bit). If you want to convert this to Q15 you take the top 16 bits, if you want Q0 you take the bottom 15 bits plus the sign bit.
The problem is that fixed point audio processing is difficult and 16-bit audio processing is extremely difficult. You need to constantly manage head room and optimize signal to noise ratio for each processing step. Fixed point filters require careful management of section ordering, section gain, section topology, noise shaping, rounding strategies, limit-cycle prevention, etc.
If you can afford it, I strongly recommend to do this in 32-bit floating point, which is much, much easier. If you can't, consider 32-bit or at least 24-bit fixed point. If that's not possible either, at the very least use double precision (fixed point) accumulators and filter state variables. If that's also not an option, you have a very rough project ahead of you.