Electronic – DIY Isolated Power Strip

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.)

Metal strips are used in swtichboards in my country because they can handle more power and have less risk of catching fire (cheaper maintenance, less human error risk). They are also cheaper to manufacture, since the strips are connected and held in place by screws and not soldering. Soldering a wire to a metal strip is not especially easy, though, and there's a risk of not doing a great job with one wire and it coming off and touching something else.

The ideal way, in my opinion, would be to use some sort of thimbles which you can hold together with nut and bolt. If you're planning on making a PCB, you could use the PCB itself to distribute power with ultra wide traces (not a good idea if you want to control heavy duty electronics like ACs or room heaters). The PCB would give you the freedom to leave large pins for the thicker mains wire to be soldered properly.

The optimal solution without a PCB or buying new components like thimbles, in my opinion, would be your option two where you use the caps as thimbles. I would advise caution, though - the outer surface of the the caps are also conducting and therefore these would have to be well secured and insulated. Remember to not skimp on the wire thickness to accomodate more wires per cap - that could lead to burning plastic, broken wires, and worst case, electrical fires. A fuse may be a good idea to include in your box, to be safe.