Electronic – Series pass transistor, LM7812

No, what you proposed won't work, and here's why.

1. 15V at 3A is 45W, not 600W. Your input power is insufficient, so adding pass transistors will do absolutely nothing for you.
2. You cannot stack 4 12V regulators on top of each other unless they are isolated. With one transformer they would not be. Youu would be shorting the 12V output to ground.
3. The 78xx series' output current is 1A max. 600W would require 12.5A, 1250% over spec.
4. Let's pretend you could get it to work and it was 70% efficient (good for a linear supply). That means you would have over 200W of heat to get rid of. That would take some enormous heatsinks.

You really need to look at some sort of offline switching regulator for that kind of power.

I suggest you do not use a 3.3V zener diode as a reference- you'll get horrible line regulation (and ripple rejection), especially with a resistor as the current source, as well as bad temperature stability.

At least use a TL431 (almost as cheap as a zener in volume) which (if you give it >1mA) will maintain a very steady voltage (nominally 2.495V) and has quite reasonable temperature stability. Your 1uF in parallel should result in unconditional stability.

An LM358 should work okay with a sufficiently low pullup resistor to allow the output to get close to the positive rail so the MOSFET can turn off. The LM358 is good for 32V, your MOSFET gate is probably not rated for 32V so that limits your maximum input unless you improve your circuit.

You've not made any attempt to deal with (frequency) compensation. At some point (probably very soon) you will find out why LDOs have problems in this area when it turns into an oscillator. You may not easily see the oscillation at the output because of the huge capacitor, so look at the op-amp output to see if the circuit is stable.

Anyway: How to test

1. Line and load regulation test for the rated input range with different loads (minimum to maximum and a few inbetween). Measure the output voltage for each.

2. Temperature stability- repeat tests at different temperatures from minimum to maximum.

3. Stability- change the load from maximum to minimum with different input voltages and observe the output behavior-looking for droop or overshoot when the load is increased or decreased suddenly.

4. Ripple rejection- apply some ripple on the input at the desired frequency and observe how much gets through to the output.

5. Drop-out- observe the output behavior as the input is increased from zero to a voltage a few volts above the output, with various loads.

You can measure quiescent current Iq as well, if you like. If you pullup resistor is very low (like 1K) you may see a significant increase in current as the op-amp rails for a high set output voltage and input slightly too low to regulate.