The tone of your question implies that you have little-to-no experience with switching power supply design.
You are going to have an incredibly difficult time if you want to make a transformer with a single primary and sixteen secondaries. The construction of a transformer is often more critical than the hard electrical/magnetic parameters (turns ratio and core material) due to their being so many degrees of freedom (leakage inductances, coupling ratios, copper loss in the windings, interwinding capacitances, etc.).
If the secondaries have to be isolated from the primary, but can be common to each other, you can go with a single secondary winding rated for all the power you need, and use point-of-load converters (bucks or synchronous bucks) to regulate each rail and provide overload protection (to keep one rail from bringing down the entire bus). You can get complete synchronous buck stages in 2mm square packages (a few external parts and you're done.)
If all 16 rails have to be isolated from each other, I'd recommend not using more than four secondaries per transformer (obviously you need four converters). You could go with a flyback converter design, which simplifies the secondaries (no filter inductors needed) and allows for output > input with galvanic isolation. There are many integrated flyback controllers on the market that contain the MOSFET and control circuitry, just wire up some feedback through an opto and away you go.
You (of course) need a properly-designed transformer, so "yes" the turns do matter as well as the actual number of turns used. The number of turns impacts the inductance, peak current and peak flux density of the transformer. A proper transformer design optimizes the number of turns to minimize core and copper losses, and requires a thorough design procedure. There is no 'magic' number, and more is not always better. For a flyback converter, there are more/different constraints, since the transformer has to be designed to store a certain amount of energy.
Your space budget is small. Forget about sinusoidal waveforms. Forget about low frequency operation. You need high-frequency conversion to minimize the space, which (in its simplest form) involves square waves. Of course, there are efficiency tradeoffs with higher frequency operation. (Space doesn't come free.)
If it's a true microphone input, then it's looking for mV signals. A "line in" port usually wants around 1 V RMS. A "speaker" output port will be low impedance and have some power capability since it's intended to drive 4 or 8 Ohm speakers, usually to at least a few Watts. Anything else is hard to guess.
Best Answer
It is done by magic ;-) Just kidding, it is not. It may look like a big deal but it's not actually. See the other answers, they are also correct.
The power from the mains is actually converted into MAGnetIC energy (do you see what I did there ;-) )and then back to electrical energy. Using very fast switches at the high voltage side the amount of energy that is transferred to the low voltage side can be precisely controlled. This conversion to magnetic energy also has the advantage that the output can be isolated from the mains supply so that you do NOT get an electrical shock when touching the output, I consider this is a very nice feature !
A reference voltage is made internally in the power supply's chip(s). Generating such a stable reference voltage is done using a bandgap circuit. The output voltage is compared to this reference voltage and adjusted so that it will be the correct voltage, this is called feedback. This feedback controls the circuit on the mains (high) voltage side and tells it to put more or less magnetic energy in the high frequency transformer depending on what the output voltage is doing. If the output voltage is too low: more ! If it is too high: less ! Simple as that. This feedback signal usually travels through an optocoupler (so using light) so that it does not need a direct wired connection to the mains side electronics and so keeping the isolation from the mains voltage side.
Due to this feedback the input voltage can vary over a wide range while the output voltage remains constant. Brilliant isn't it ? ;-)