First off, congrats for you working on this. Good to see people taking an interest in really learning what is going on rather than just changing parts. And you have a scope! A+
Quench is giving you some really sound advice. Kudo's to Quench too.
So I'm a little late to the party here... I got a little confused reading the thread, seems like we are jumping back and forth between the 12V, the 5V, and boards in games. So apologies if I missed something. I likely did. My primary job isn't designing power supplies, but I do it away since we tend to add regulators to our own boards to ensure a good, solid supply. Not an expert by any means tho. The LM323's are pretty much standalone parts. A SinpIt from the TI data sheet is below. Pay attention to the capacitor on the input. Typically what is important besides being at least a 0.1pF, is the cap has a low ESR. While I'll add a large cap in my designs, sometimes large, cheap capacitors aren't low ESR, at least at the frequency frequencies of interest. It not uncommon to add, say a decent 0.1uF right at the input to the regulator in addition to the larger cap. If I'm not mistaken, it also keeps the internal loop of the regulator stable. 15,000uF is a fair amount of capacitance At the input. Keep an eye on inrush current charging that cap (not sure if you know how to do this). An inductor there might help. The ground on the LM323...in your case, that's the '-' line coming from the bridge and cap. Simply wire it like below. The SnipIt shows a LM123 - that's just a different flavor of the LM323. It's wired the same. If everything works w/o the 0.1uF, then don't sweat it.
Q1: That circuit should work, but it's pretty barebones. The zener (ZD1) sets a reference voltage on the base of the transistor, and the emitter is simply a diode drop (0.7V) below that then, but it can supply high current. If the base current isn't adequate, then potentially it won't regulate under load. This is defined by the 'beta' of the transistor. R1 sets the current thru the zener and base, and C2 is just to filter off any noise on the base. If C2 wasn't there, any ripple on the base would also show up on the output (i.e. with ZR1 and C2, you should have a nice quiet reference, therefore the emitter would follow suit). Honestly, the Q1 stuff...just pitch it and put in a linear regulator, like a LM137 (if 1.5A is enough). There's hundreds to choose from, and all would be far better than the barebones +12V regulator you have.
Since it appears you have a knack to learn the basic details, I highly encourage you to download LTSpice (free) which is a simulation tool by Linear Tech (since bought by Analog Devices). They make hundreds of regulators for the industry...you can simulate all day long and not burn up a single part!
Here's a link for LTSpice: https://www.analog.com/en/design-center/design-tools-and-calculators/ltspice-simulator.html It's rather impressive for free. I even use it despite having the expensive simulators to use since the library of their parts are included for free, and the models have been excellent so far. I'll help you get going w/it via PM's, if needed. Or you can post here, but I didn't want to derail your thread.
There's a lot I skipped or glossed over in my comments, holler if needed and I can explain (maybe).