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.)

You should be able to control inrush current with eg a series MOSFET with a network on the gate which controls the rate of rise of gate voltage. Their DVCL (Delta Voltage Current Limiter) controls the maximum rate of input voltage slew (dV) BUT by simply applying a slowly rising voltage to a MOSFET gate you will produce a "resistor" of decreasing value as the gate voltage rises. The characteristic will be non linear and voltage rise will be nowhere near linear as they show BUT will be able to be made as slow as desired.

The diagram below shows their claimed DVCL results - see their video for scales.
Yellow = Vin to DVCL and Green = Vout to circuit proper.
Blue = current.

If you place say a P Channel MOSFET in the +ve lead (source to Vin, drain to load, with a resistor from dare to ground (-ve) and a capacitor from the gate to ground then:
As gate voltage rises exponentially, as it rises above Vgs_th the MOSFET will start to conduct and the MOSFET will be fully on typically when Vgs is in the range 1V to about 5V above Vgsth. A little playing or some simple caluclation will allow the correct RC time constant to be calculated.

Excellent Motorola application note AN1542 - active Inrush Current Limiting using MOSFETS expands on this basic concept and provides much detail and background. Their not apprecioably more complex fig 5A provides an example real world solution.

The MOSFET will obviously need adequate voltage and current ratings and a suitable Vgsth. Rdson needs to be suitably low for your application.

You note that Iin = 3A steady and 8A peak and Vin = 24V.
Effective load resistance at 8A = V/I = 24/8 = 3 ohms.
Loss in MOSFET or in load = I^2 x R, so a MOSFET with Rdson of 0.3 Ohm will dissipate about 10% of the power used. But at 30 milliOhm Rdson dissipation will be 1% of power used and lower Rdson is better.

Circuit power at 8A, 24V =~ 200 Watt so peak power dissipation at Rdson = 30 milliOhm ~+ 2 Watts. See MOSFET datasheet for acceptable pulse-time ratings to decide if a heatsink is needed. A sensible TO220 device should be safe with startup times of around 1 second or faster and no heatsink. TO89 / SOT223 probably still OK especially if a little PCB copper is available for heatsinking. The excessively enthused could probably design a low Rdson SOT23 MOSFET into this circuit with due care.