Truck will not start, unless battery Charger is hooked up. What is wrong?

Start by checking the fuel pressure and ignition switch

You have way too many things to address here. There has obviously been hands in there modifying things that we have no way of knowing or figuring out. The engine can't run without a mass air flow sensor signal. Hooking "extra" wires to the battery invites a drain from who-knows-what. No one would just assume any leftover wires go to the battery. You've also started randomly replacing unrelated parts which adds more variables to the original problem. What is a "heat sensing unit"? Do you mean the coolant temperature sensor? There could be two of them. Intake air temperature sensor? What's an "ignition sensor"? Do you mean the crankshaft position sensor? Camshaft position sensor? Distributor pickup assembly?

Cranking for more than few seconds suggests a fuel pump of fuel supply problem. That would appear to be completely unrelated to most of the other problems.

Probably the best place to start is by finding a manufacturer's service manual, (not a Haynes or Chiltons), and using the wiring diagram to determine which wires go where. Along with melted wires, you'll have to determine what's at the other end and if applying battery voltage would have damaged anything else. When the person you bought the truck from disconnected the mass air flow sensor and left it that way, that shows they didn't know what they were doing or how to fix it properly. Some engines will run off the MAP sensor as a backup, but only Chrysler has been able to make an engine run properly without a mass air flow sensor. Even if you get the engine running, it won't run well until you correct the mass air flow wiring. Look for a service manual on eBay or at an old car show swap meet. Some vendors sell nothing but old manuals.

To add to the misery, it sounds like you have a generator / charging system problem if you need a battery charger. I can help with that diagnosis once the engine is running. All you'll need is an inexpensive digital voltmeter.

I think I'd start by monitoring fuel pressure and / or squirting a little fuel into the throttle body. Starting fluid would work too. If it starts right up on starting fluid without the battery charger, you'll know it's not a spark-related problem, ... Well, ... Maybe.

Fuel pressure can be a little tricky to analyze. On some cars, the engine will run if the pressure is just close. On some, GMs for example, the pressure can't be off by more than a pound or two. Watch the pressure before and after the charger is turned on. If it goes up when the charger is turned on, that would point to a pump problem but that would definitely not be my first guess.

Something else you might consider is whether there is a resistor in series with the ignition coil. If there is, that is to limit voltage to and current through the coil. Since battery voltage is already down a couple of volts during cranking, the additional drop across that resistor would result in a very weak spark, so they typically bypass it during cranking. Chrysler used to do that with an extra terminal on the ignition switch. Ford and GM used to do that with a tap on their starter solenoids. Perhaps that is related to one of those two extra wires. That bypass circuit isn't used once the engine is running. If that circuit is needed but not hooked up, the higher voltage from the battery charger might be helping to increase the voltage to the coil to the point it develops a sufficient spark. Don't overlook something stupid like a spark plug gap that's too big and the spark can't jump across it.

If you don't see any increase in fuel pressure when you turn the charger on, look into a weak spark issue. The confusing part is starting fluid might fire easier than gasoline giving the false impression that fuel is the issue. If it starts on starting fluid without the charger but not with gasoline sprayed into the throttle body, I'd suspect weak spark.