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.)
The point of a isolation transformer is to break any common mode connection to the rest of the world. This therefore allows one connection to the rest of the world that won't draw any current. Usually you try to not have any such connection, so that a accidental connection (you touching a live wire with one hand while grabbing a metal drain pipe with the other, for example) is the first connection and therefore safe.
A common mistake people make when using a isolation tranformer is to forget that connecting a oscilloscope usually means connection some point to real earth ground. That by itself is OK since it is only a single connection. However, now the circuit is no longer isolated and significant current can flow between some part of the circuit and a metal pipe, for example.
Best Answer
Because the mains supply is very unpredictable, and can do all sorts of things outside its nominal specification, which might damage components or at least break the nominal design assumptions. A non-isolated design also has all its voltages referenced to one of the mains conductors, which might or might not have a useful/safe relationship to other potentials in your environment (like earth/ground, for example).
If the only stuff on the low-voltage side is inaccessible electronics, then non-isolated supplies are fine - they tend to much be cheaper/simpler than isolated supplies, and lots of household equipment uses them. Even things like televisions used to work like this, if you go right back to before the time when they had external video/audio connections. The antenna connection was the only external socket, and that was capacitor-isolated.
If a human being or 3rd party piece of equipment needs to interconnect with the low-voltage side of your design, then an isolated supply both gives you a clear barrier across which dangerous voltages won't pass, even in the case of component failure, and it means your circuit is now 'floating' relative to the mains. In turn, that means you can arrange for all the electronics to operate near ground potential, with all your interconnected equipment having at least roughly the same voltage reference to work from.