Why in the world would I be seeing 10+ more volts here than incoming? And how?
It's because at the fuse, you're measuring rectified but not smoothed DC (basically same as AC on the transformer, minus bridge drop), smoothed wave on C8 will be √2 times higher (minus D6, Q1 drop).
Also check R9, it have seen some overload, may be damaged. minkok is absolutely right, should be the zeners, maybe a cold joint.
That wave is weird. Where do exactly you measured it? For a transformer output it's much like a DC wave with ripple, for board DC supply input it has too much of ripple. Maybe D6 is busted?
U17 is clearly not amplifying, though there should be fixed x28 gain. Maybe it's the DAC. Check U11:1, there should be 0V in calibration and with C245.
Why it should be a problem? Just to clarify, if the main supply is SMPS, aux supply is also SMPS, if the main supply is a mains transformer, aux supply is a separate (floating) winding in it. Supplies GNDs, of course, should be tied in one and only one point closest to the power input. Original JBC stations is done in this particular manner.
For the driver? It doesn't really matter as long as the driver output voltage is enough to fully open the MOSFETs (and not exceed maximum driver supply, obviously). Zeners left there to act as suppressors to protect the driver and the converter. It's not about the price. Fewer parts mean less hassle to put it all together, less space required. If it would reduce the BOM, even better.
Those parts are required for a higher voltage input and it's a robust design. If input is limited to nominal 12V, they can be dropped to simplify the power stage. Nominal quiescent current of TC4428 with 3V input - 4.5mA, which means voltage drop of around 1.5V on R20. When nominal input voltage is 24V, it's not a big deal, but with the input voltage of 12V this is significant drop. Though, it's proved to be working fine on 12V supplies in C210 configurations. 330 Ohms is really huge resistance for that application and it's not limiting driver current, since it is directly provided by C19. I believe it's there for decoupling.
If aux supply is 10~15VDC, Q13 also can be removed and bridged, D14 removed and bridged, R19 removed. R20 has a rather large resistance, something like 22~47 Ohms would be enough to suppress driver noise.
What stabilization circuit? D6,R2,D5,Q1 is removed, Q1 collector-emiter bridged, D5 bridged, aux supply connected to C22. That's all. AC rectifier may be designed as a patch board. There will be some drop on Q13 circuit, but it should produce enough voltage for lower mosfets, even 8V resulting gate voltage would be enough.
It shouldn't cause any problems, there is DC-DC converter after it anyway. Q5 circuit is 6v, plenty of headroom with 12V input. Yes, when using separate winding, bridge rectifier and cap is required, a bit ugly, but it won't take much space and will be fairly reliable.
admaverick, minkok, it's much easier to just implement auxiliary DC supply, rather than messing with values and dropping 30-odd Volts on Q1. If I was going to redesign the power board to support higher voltage irons, I'd use 12V supply and get rid of Q1 and Q13 circuits all along. It could be a separate 3W AC-DC supply, like IRM-03-12 or separate 9VAC transformer winding with it's own rectifier and filtering. If you have a toroidal transformer, just wind a turn on top and measure the voltage on it, then wind the remaining.