Yes, under the right circumstances but they aren't random. It won't throw the code until it is running and the reading from the sensors aren't in alignment with each other. Low cylinder pressure can change rpm, manifold pressure, TPS, CPS, etc. to have misaligned readings from where it expects it to be which will throw codes. However, 120psi is not really low for a SC motor and I would expect it to run just fine with your values.
The SC is geared up so that it spins around 35k+ max rpms so that when you're turning the SC the flywheel isn't moving that far, around a 5 to 1 ration of engine rpms. If you continue to turn the sc more than 1 revolution without releasing pressure, it will eventually slip forward instead of backwards. You should always turn the SC nut CCW since it is a left-handed nut and you don't want the impeller nut to loosen.
The simplest way to turn the engine over is to turn the drive shaft or bump the starter.
The engine will not keep itself from firing on a single cylinder as it has no idea what the compression is on any cylinder and will always maintain spark and fuel to each cylinder equally meaning it won't sense a bad cylinder and change spark or fuel delivery.
Does the engine run? Does it misfire? Does it rev up? In limp mode? Need a little more info on the actual running characteristics to advise what the next steps are.
All of that makes plenty of sense to me.
To answer you questions: Yes, it will run, but it runs rough and has a hard time staying running. It is not in limp mode, at least not to my knowledge. I have yet to be in the water with it, so I'm not 100% sure on that. Yes, It revs up.
To be clear, when I say random, I just mean that I don't necessarily get the same codes every time. At first, it was the crank position code. Then it was the cam code and the blow by valve code, then a couple others. They seem to come and go. The check engine and flashing red light is consistent.
I decided to order a Spline/Impeller tool so I could properly spin the motor. It came today, so I took the super charger off and pulled the drive shaft so I could manipulate the engine more precisely while doing the leak down test. Cylinders 1&3 tested nearly perfect. The middle cylinder(#2) tested right in between good and moderate(39 or 40%). It leaks out both the exhaust and intake valves, as well as into the crank case. The majority of the air seemed to be leaking out of the exhaust valves. I'm hoping that means I can get by with just yanking the head, instead of the whole engine.
I also think there is enough compression and cylinder health for this thing to run on all cylinders. The cylinder not firing at all is surely being caused by something else. I've moved around the coils and plugs, and even put new plugs in it, and the misfire didn't move cylinders. I also pulled the fuel rail, and the injectors all seem to be spraying the same.
One other thing I noticed while doing all this, and I mentioned it before, is that when I'm turning the engine by hand.......it will spin easily in parts of the rotation, then it gets harder to spin, like it is binding or pushing through some compression, then it will get past that point and spin easily again for about a half rotation. I have no clue if that is normal or not. In my mind, it should rotate freely and smoothly throughout the whole rotation, especially since the plugs were out, but I really don't know.

It seems to turn over just fine with the starter.
So, where do I go from here is the question? How can I rule out something electrical or ECM related? I'm fairly mechanically inclined, so the thought of pulling the engine and rebuilding it is somehow less intimidating to me than troubleshooting the electronics. Electricity is like wizardry to me
ETA.....And whoever posted on the internet to put a wooden dowel down the spark plug hole to tell when the cylinder is at TDC is giving very bad advice. It snapped the dowel right in half and I spent several hours fishing out bits and pieces of wooden dowel rod.
