PCM replaced and Fuel injector not working after reversing the battery connection?

Hello, with all the injectors unplugged, key off, none of the injector control wires should have any continuity to ground at all. If they do, unplug the PCM connector and retest the injector control circuits for shorts to ground with all the injectors unplugged. There should be no paths to ground at all. I'll post a wiring diagram for you, where did you get the new PCM?
Are you sure you're testing the correct wire, it sounds like you're checking continuity on the power feed with the one injector plugged in.

With a battery reverse hook up, any number of different modules could have been damaged, which is usually the case. I would suspect a module issue more than a wiring problem. But determining what is failing just before the stall out can be tricky. It might be losing fuel pressure or spark causing the stall. Looking through the wiring diagrams, one model has a fuel pump resistor which varies the fuel pump speed.
Is it setting any codes in any modules due to the stall out?

And you were getting the same P0606 code prior to the battery issue I assume? That code is labeled as ECM/PCM Processor Error which if you have replaced the PCM already is being caused by some other issue. Another potential cause stated is "Ignition Switch Terminal (PCM Terminal 1AY) Circuit Shorted to Power" which is a bit strange because its a 12volt feed from ENGINE FUSE 20Amp to the ignition coils and other components so being shorted to power as a possibility doesn't make much sense. I do see a PCM Ground pin next to the 1AY pin, so I'll look into this code some more, but check the PCM connectors for any corrosion deep inside the connectors or possible spread connector pins,
A spread connector pin would make more sense to me being that this issue happens shortly after start up, If there's a PCM connector pin that's slightly spread, heat after start up is going to cause that pin to lose contact pretty quick.
Does the vehicle start right back up, or do you need to wait a certain amount of time?

Where did the new PCM come from?

Okay, first thing to check is this fuse, even if it looks okay, don't take that for granted, swap it out with a different 20-Amp fuse and check the fuse location for any corrosion or spread pins, look closely at the fuse as well, does it look dark on the pins, like it has been over heating somewhat. This code can be misinterpreted by the PCM as an internal fault because of a lack of voltage. Thats the first thing to check,
It's in the passenger side junction box.

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Here is a very good video for you to watch, if you're having the same issues with different PCMs then it's not the PCM. It's either going to be a battery, battery connection, or wiring to the PCM. Low voltage fluctuations can cause these strange PCM error codes, the PCM can't function normal when voltage drops out momentarily or a possible ignition coil is jumping to Ground and spiking the PCM, causing it to reset, shutdown, and it requires an engine restart. But watch this video, it will help you out a lot.

https://www.youtube.com/watch?v=Aj7SAn2F3V8

I'm glad to help if I can.

There is a variable intake manifold variable swirl shutter valve which will vary the air turbulence coming into the intake manifold under different engine loads, the idea is to improve the horsepower under different engine loads. This is the TSB about it sticking in colder weather but will set a code P2006. There are some PCM updates having to do with that issue. I'm trying to find a decent picture of it, it looks like there is a rod that moves the shutter valve (2nd diagram). But usually what happens with these variable intake runner type designs, is because they are made of plastic like the intake manifold is, they eventually warp due to heat, and that causes them to stick in one position or another.
I don't know if it would affect the manifold enough to stall the engine or not. Sometimes I find that they are causing a vacuum leak, because the rod is external to the intake manifold, I find some leaking when doing a smoke test on the intake manifold.

If they can show you exactly why they think the intake manifold is causing the stall, that's a different story, but does the engine stall out even if try to give it some throttle?

In a case like this I would try to watch for whatever the engine is losing, such a fuel pressure, spark, or some voltage loss to the electrical system, instead of just replacing parts and hoping that solves the issue. That code (P0606) is an indication of what might really be happening. Does that code still set after the stall out? Because you could setup a multimeter somewhere and see if a voltage loss is causing the stall.

This is the circuit going to the PCM that's setting the P0606 code. Pin 1AY Green/Blue wire, it comes from that Engine Fuse 20Amp. If this circuit loses voltage, it throws off the PCMs operation. So, you could monitor that wire, it should be 12v while running. And see if it drops out during the stall. The fact that you can restart the engine right away tells me the PCM is most likely glitching out, I would not like to see you replace the intake manifold and still have the same issue.

Is this a California Emissions vehicle or Federal Emissions? It should say on the tag under the hood. There's some differences with the PCM wiring depending on the emissions specs.

The first diagram is the California emissions with auto trans, with circuit 1AY being fed power from the Ignition Switch, the 2nd diagram is non Cali emissions auto trans with circuit 1AY is the Drive by Wire relay control wire, so it will not read 12volts when the ignition is on, whereas the Cali emissions vehicle will have 12volts on 1AY when the ignition switch is On. So that's the difference in circuit design.

These are the two different PCM connectors depending on the emissions regulations, and the pin 1AY to monitor the voltage level during the stall/fault condition. The 1AY pin is located in the same location either way with the green/blue wire.