If you can find a shop w/the proper scan tool, they can tell you using the real time fuel trim data if the engine is otherwise firing ok, excepting your cold start problem. If the cold start is the only time you get misfires, then that is very often caused b/c the fuel rail doesn’t remain pressurized like it should when you turn the key off. The fuel rail is sort of like a balloon, you blow into it until it gets pressurized, then you tie it off, and it is supposed to stay pressurized. But sometimes – just like balloons – they can get leaks. The fuel rail can leak three common ways, one is back into the tank through the check valve, one is through the fuel pressure regulator, and one is through one of the injectors. There’s a fuel pressure test called a pressure leak down test that would confirm if the fuel rail is losing pressure. If it is losing pressure, this symptom would be the result, and then its just a matter of figuring out which of the 3 ways it is leaking.
These are both random misfire codes, the 300 for multiple cylinders, the 302 for cylinder #2.
The trick is in figuring out the “why”. Misfires generally place the ignition system as suspect, however in this case it’s possible that difficulty getting the fuel pressure up may be the cause. Failure to combust creates a different spark pulse, as can be seen on an oscilloscope. That defective pulse triggers the fault code.
I’m still going to go with the fuel line depressurization at the moment. New information could cause me to rethink that. But keep trying the trick you tried. Or, as George suggested (a suggestion I like), the fuel rail can be tested for pressure leakdown. It isn’t a difficult test. It only involves a T-fitting and a fuel pressure gage. Any shop can do it for you.
I’m still sticking with the valves
My reasoning is this . . . a bad fuel pump check valve does NOT explain the very specific P0302
In my opinion, there is something going on with that #2 cylinder
For OP’s sake, I hope I’m wrong
to add to it, both times the shop had the car, they reset the codes. it tends to fluctuate with codes. It always shows the p0300, but sometimes is 0302, or 05, or 06, or some combination. The constant is the p0300
@gimbit
Thank you for the additional information
Has anybody “smoke tested” the intake system yet?
Failed intake gaskets tend to swell up and seal better as the engine warms up
I’ve personally seen intake gaskets cause misfire codes and rough running
I know they checked the head gasket, that was fine. Was told they replaced a gasket. Not sure what they replaced. Know they also replaced plugs, coil, etc. Apologies, am dealing with very sick child, so my thoughts are not complete right now. Why don’t they just drop a new engine in it and call it a day? 
It is difficult for a shop to diagnose a cold start problem on a hot engine. Did you leave the vehicle for a few days so they could experience the problem?
The cold start misfires that occurs on 3.7L/4.7L engines with a fuel system defect because when the vehicle is parked with a hot engine and fuel pressure is lost, the fuel in the fuel rail will boil. This will push the fuel back to the tank leaving the fuel rail empty. The fuel rail and engine are mounted on an angle, uphill. The next time the engine is started the liquid fuel will enter the fuel rail and the vapor will be pushed to the front cylinders, causing cylinder #2 misfire.
Most engines are relatively level and with this problem the vapors are pushed through every injector resulting in a long crank time.
I used the term “fuel system defect” because this must be diagnosed. I have found fuel pumps that fail to hold pressure but a leaking injector is a possibility.
Hello. Last time they had my car for 4 days. Am about to start car and will try the trick again. Will post what happens. I appreciate all the info!
i have noticed as i read those posts there are many reason as to why so i just decided not to read any longer and checked my problem myself and went on to go for the easiest start.i had the same problem untill just now and so i knew i checked all my oil and water levels and cold level and brakes but still the engine light was still on untill now. what did i do? first thing i removed fuses and all fuses were fine but i noticed they were very hard to pull and was quite Rusty like, not exactly rust but some kind of rust i guess. so i cleaned the fuses connection and put fuses back in went inside and put the keys on untill the engine light flashed 3 times and then stayed on and then i started the Jeep and the engine light problem ended.voila.
Sounds like by removing the fuses you cleared the computers memory, now you won’t know what fault was stored until the next time the check engine light comes on, which shouldn’t be long.