Good work, Bernie! The coolant temp sensor isn't something I'd have suspected when the engine starts OK cold, but results don't lie.
I can't find this P0256 code. Probably because it's a diesel code of some sort. I can't explain that one... unless the code reader isn't compatible with your truck. When the scan tool and the vehicle don't connect odd codes like this can pop up sometimes. Don't worry about this one; it probably isn't real.
David brought up a valid point. If you can pretty much verify that you have fuel pressure when the engine is acting up, it just might be a problem with rust collection in the injector pickup screens. It's not iron oxide rust, but some sort of stainless steel reactant. It's uncommon but does happen.
The source is your fuel pump (if it's not magnetic). Fuel passes through a filter/ regulator assembly in the top of the pump assembly that's built of some sort of stainless steel (or similar) material. The stuff we're talking about is shed from the inner walls of the filter/ regulator can and it happens AFTER the filter media, so there's nothing to stop it from heading up to the engine.
Verification goes like David said... remove the rail and tap out the injector screens to see what falls out.
The one thing that bothers me about this is that you have a 4-mile honeymoon each time you start the truck where it runs OK. Restricted filter screens usually STAY restricted and cause the engine to run worse when cold. Tough call, especially with the plastic fuel rail...
Since you've already mastered the art of hotwiring the fuel pump, maybe a fuel volume test would be in order. Get yourself a 5-gallon can, a lenth of 3/8" fuel line and a fire extinguisher LOL. No, really get the extinguisher. Gas is dangerous stuff.
Jumpering on the same side of the truck as where you will be dispensing fuel puts these two components too close together for comfort. Be very careful to avoid stray sparks when liquid fuel is present from fuel line disconnection. Let the fuel evaporate if possbile before firing up the pump.
Then just run the pump and let the fuel fill the gas can. Time the process and watch the fuel flow to see if it markedly peters out after a short while.
Be sure the gas can is clean inside so that any particulates you may accumulate can be blamed on the correct source... your fuel system.
One disclaimer... Some fuel pump modules just naturally pick up fuel poorly at fuel levels below 1/2. You may have to add fuel to get it to 3/4 for this test to be meaningful if it begins sputtering at the discharge hose after 20 seconds.
Last, I thank David (ProAnswerMan) for his interest!
Talk later, Ed