After replacing the transmission solenoid pack, transmission is not shifting properly and codes P0736, P1603 and U0101

From the description you really need a dealer level scan tool to watch the various sensor signals on the transmission. The P0882 showed low power to the transmission control unit, which could be a wiring short, but it could also be a failed PCM. The P1603 and U0101 are codes relating to the transmission controller not being able to communicate with the PCM or internal faults of the PCM. So that also points to it either having no power or internal failure. The P0736 would make sense in that the transmission section of the PCM controls all of the shifting and gear changes using the pressure in the transmission. If it has a power issue or internal failure it would set codes like that. As to needing a dealer, not true, you just need a better shop that can actually do diagnostic work.
The other item is the leak into the connector, those can cause all kinds of issues because it will short wires that should never connect and send bad info or voltages into the PCM and cause it to fail.
I would check the powers and grounds for the PCM as Chrysler has had a lot of issues with both of those and either can cause problems. As it starts and runs, I suspect it is more on the ground side, all four of the transmission related grounds go to the same point as shown in the third image, those multi wire connections like to fail.

https://www.2carpros.com/articles/how-to-check-wiring

Was the pack a new pack or used? Either way it could be a bad solenoid, or something got into the passages. I would start by checking the connectors pins over real well, it's possible that the one for reverse is loose or corroded and not making contact, or the one for the speed sensor has an issue and is not seeing the correct signal when it goes into reverse. Does it try to move at all in reverse? Even loading the engine would be a sign that it isn't fully internal, but if you can shift into reverse and nothing happens then I would check the wiring first, a somewhat easy test would be to use the autostick and shift it down to first gear, does it move then? The same solenoid controls both low and reverse, just applies a different clutch to give you reverse. So, if it starts out in first and shifts up through each gear manually the solenoid itself likely isn't the issue.

That description sounds like the solenoid isn't working, the kick in drive could be that it isn't engaging low gear and instead is starting out in second gear. With the old pack you had the error, but it worked in all gears, correct? If so, I would take a meter and test the solenoids for resistance, that can be done from the wiring connector. Take a meter set to resistance and connect one terminal to pin 4 on the transmission connector, then measure the resistance between it and pins 4, 5, 6 and 7 in turn. The readings at each one should be almost exactly the same. If that checks out, then it's likely that something got into the passage for reverse/low. If that is the case, then about the only way to get it out would be pulling the transmission pan and pulling the valve body apart to remove whatever is blocking the flow of fluid.
Removing it isn't really hard but it's messy.
First thing is to remove the battery and tray so you can get to the manual shift lever on the transmission. Make sure it is in low when you do that as it makes it easier.
Next you remove the pan and filter, next you remove the bolts that holt the valve body to the case. Now you need to grab the parking roller from the guide. Then remove the valve body. Tag the two accumulators that are in the case.
Before doing this though you need to be sure you have a scan tool available to do the relearn procedure. I might even try a relearn before you start pulling parts.

It should go as low as one normally when it is selected manually. Relearn should be performed after most transmission repairs. You need a scan tool that can do bidirectional control to do it and that depends a lot on the shop's tools.

There is a roller that sets at the end of the park lever inside the transmission, image attached. You need to remove it to get the lever out and not let it fall into the case and cause problems.

Transmission should be in low to get it loose, but it doesn't hurt to move the linkage to get it to drop free. I would be sure to put a big sheet under the car if you are doing this with the transmission in the car as the removal of the valve body will drop the check balls free and they love to hide. I would also suggest getting the one-year subscription to Alldata DIY for that vehicle, it has the full-service info for this procedure as well as showing where each ball goes. Runs about $20.00 but only covers one vehicle, you can print or same images in higher resolution as well.

Hmm Mitchell should have the same info I posted above; they both mostly copy the OE information. We use both but I prefer Alldata as I'm used to using it, Mitchell tends to move things around, so it takes more to dig them out. Take a look under transmission, procedures, removing valve body. As for stripped bolts, best solution would be to use a Heli coil kit to repair threads and for replacement bolts I tend to hit you pull its or just order the bolts through a dealer if they are something odd. You might find some metric bolts in places like Lowes or a good hardware store or McMaster-Carr but they can take a while to find on their site.

So, you're the one who is flooding my place huh, Thanks a lot. LOL. No rush, we'll be here, I know what it's like to have an open roof shop.

Good luck, Take a snorkel and swim fins. Hopefully, whatever is blocking the flow will show up in one of the valves. Trying to get in deeper isn't fun. I think they do show the fluid paths on there as well so you might be able to use some air pressure on the passages to be sure they are clear.

As long as the lever moves it will shift, but it's possible that whatever is causing the fluid issue is in the main control spool that the lever moves. I'm hoping that you pull the VB off and find a piece of gasket material in one of the check balls or sticking a piston open or closed. Then you can reverse the process and have a working vehicle again. Just don't be like this person.