We had someone tow a 6.0 in for a no start. The first thing I saw was a spare tire on the right front and knew the switch was tripped. I've even kicked the kick panel on a buddys truck to screw with him. Sometimes the switch is too sensitive.
You have hydroboost. It looks more like the master cylinder is leaking. Check to see what fluid is low, brake fluid or power steering fluid, that will tell you what is leaking. If the booster is leaking it would be red because it works off the power steering pump.
It wouldn't hurt to look but most of the cavitation problems I've seen are on the 450 and 550s due to the lower gear ratio. If you pull the pump just look at the cover behind the pump to see if there is a problem, it will be very easy to tell if there is.
Some pressure switches can't hold the AMP draw of a compressor and need a relay wired in. If your switch can't handle the power it will weld the leads together and be constant. You may want to test the switch and replace if bad and wire a relay in with it.
You may have to bleed the high pressure system. I've seen this on a couple of trucks. Just loosen the driverside line feeding the rail and turn it over until you see no more air bubbles then tighten it back up.
The genera rule is 400deg. but 300deg. is being super safe. Some people do time to suit their needs because it doesn't take these trucks long to cool down.
Shift linkage at the trans, pry them apart, it's a ball and socket. The brake cable is a PITA, I pull the cable to get some slack and use vice grips to keep it from going back in. Then use an air hammer to push one of the ends towards the center of the clip.
Like webb said, sounds like lifter is collapsing when cold then pumping up when starting. The only way to get to the lifters is to take the heads off. If you blow the HG and rebuild or if the problem gets worse, time to replace the lifters.
When they put in the updated hose, they can see the EGR cooler pipe that's behind it. Sometimes that O-ring will leak, bur usually after it's been messed with. Most likely the updated lower hose will take care of your problem.