Does the cooling fan work? If you remove the coolant fill cap and let the vehicle run for a bit, do you see a constant flow of air bubbles?
I was just reading through the posts and noticed that it was determined that a dye test was done.
I would recommend a combustion gas analyzer be put on the radiator filler neck when the engine is hot. This tester using "Block Test Fluid" that is blue in color. It is usually put into a tool with one or two chambers and can either have a bulb on one end that the tech can squeeze or it can hook to a vacuum source.
If the the blue fluid turns yellow once the engine starts to overheat, then it is most likely the head gasket that is failing. It can be hard to duplicate because the engine has to be very hot for the combustion gas to start to leak, but by the time it starts leaking, it is already so hot that it will start to spew out of the radiator neck. The tech has to be quick on the draw to get a good and accurate sample.
I've replaced alot of headgaskets on these engines over the years, so I really think this is where this is heading.
Sorry I don't have better news for you.
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It probably is causing the no heat and yes it could cause the check engine light to come on if there were coolant getting into the cylinder and causing a misfire, or contaminated the O2 sensor or even if the temp reading was too high it could set a trouble code that would turn the light on
You are very welcome.
I hope everything turns out ok.