Offline Flyback DC-DC converter using NE555, Works in LTSpice, will it work in a real world application?
The NE555 is IMO totally unsuitable for such a design as the pulse width of the first pulse is invariably much longer than any subsequent pulses. This is due to the fact that the timing capacitor is charged and discharged between 1/3 and 2/3 of the supply ...except for the first time you charge it, and you always start from zero charge on the capacitor. So the first pulse is from 0 - 2/3.
This makes soft start almost impossible to engineer, and can result in large initial currents since you don't know when in the AC cycle you will first make contact.  

I see no attempt here to make the supply zero crossing startup and your (I assume an attempt) at soft start looks quite flawed. Particularly the drive for Q1, Q7 and the fact that the C8 charge dump is not AC synchronized.
The first pulse from U1 will be much longer because of the ramp on C5 continuously raising the C6 aiming voltage and the complications with exactly what voltage it will get to. When your charge dump (Q1/Q2) turns on there is uncontrolled discharge of C8 through Q1/Q2 which I don't think you intended.  

If I were to test this I'd be using a transformer as isolation and start with perhaps a 50 V RMS output. 

Additional Comments 4/22 circuit:


Pre-Reg shutdown. Overcurrent base drive from Q31 driving Q30. Need a base resistor.
Pre-Regulator. Do you really need 9900 uF capacitance?
Need a 270 Ohm resistor between C9 and Q3 Collector pulldown to lower Q3 current discharging C9.
Zero-Crossing. Do you really need Pre_Charge since it's derived from Pre_Supply?
PFC-Startup Controller and Flyback Startup-Controller. Need base resistor in Q2/4. Consider when latch turns on the base current is limited only by Q18/20 Beta. 
PFC Softstart and Transformer Softstart. Clamping U2_out and U4_out appears faulty. You are trying to hold the amplitude of the signal low, but with insufficient current limiting. 
Positive voltage regulator. If I read correctly Pre_Charge will be about 17-18 V, so Positive_Voltage will not turn on. The voltage for the two BE's, 3 diodes and a zener come to about 20 V, but if Supply goes over 20 V then there is an uncontrolled current path. Perhaps you could remove a diode and replace with a resistor and still get the correct operation.