what happens if the EGR valve is taken out and welded shut?
Would it throw a code?
Would this eliminate any of the Coolant problems?
Your coolant problems will NOT be resolved by simply getting rid of the EGR valve. I don't think that welding the EGR valve shut is a good idea - but someone else can answer that aspect of your question for you.
In terms of the Coolant Problems - the valve itself has very little to do with coolant problems (some, but not much). Mostly, the coolant problems come from two sources: The EGR cooler and the Engine Oil Cooler.
The EGR Cooler takes hot exhaust gasses and runs them through some channels while allowing coolant to run on the outside of those channels. The two run next to each other without mixing together. However, when this EGR cooler fails (The Ford ones always do... its just a matter of time!) then coolant can get into the exhaust. This can allow you to lose coolant without it dripping (its like a leak but you never find it dripping in your driveway) and this loss of coolant can easily allow you to overheat.
The Engine Oil Cooler often gets plugged up and restricted. The passageways through this cooler are small/narrow - about the thickness of a penny (more or less). So any debris, sand or even minerals from your coolant that get stuck in this cooler will restrict the flow of coolant through it. That in itself is bad, but it has a down-stream effect as well. You see, your coolant flows from the radiator through the engine oil cooler and then into your EGR cooler. If the flow of coolant is restricted into your engine cooler, then its going to be restricted into your EGR cooler as well.
This restriction means that the coolant is not flowing fast enough through your EGR cooler, which equates to that coolant absorbing more heat than it should. The can/does lead to flash-boiling of the coolant and puking of coolant out of the degass bottle (your coolant reservoir).
So, as you can see, even if you DID cap off your EGR valve, all of this exhaust/coolant interraction is still going to happen within your truck.
My suggestion for coolant/exhaust/EGR concerns is our Upgraded EGR Cooler. You can find it at www.ntshoptools.com