Did you ever get the engine model and date code? It should be stamped into the valve cover. If hard to read, get some "grime" on your finger and rub it across it. The grime will fill in the letters and give you contrast. Also can take a photo of it, if hard to read.
There are several parts (fan included) that require the date code to get the correct fan.
Since your exhaust is getting superhot, it seems you are running lean... Popping and sputtering could be an ignition issue on top of this. One thing I have seen is the ignition switch (or connections) going bad and intermittently killing the coil.
Spark related, I would pull the plug and short the metal base of the plug to the engine, then crank it. If you get spark your ignition system works to provide spark. Since it starts, this is probably ok. To rule out other issues "killing" spark intermittently, pull the kill wire from the coil (you will have to remove the air cover). Start it and see if it runs ok. If so, then something is killing the coil (ign switch, safety switches etc.). NOTE: I have seen spark look "ok" with a spark checker and have this issue. You could kind of see the spark "stutter" but it was still sparking. You can kill the engine by pulling the plug wire (using insulated pliers) or short the coil kill terminal to the frame of the coil with a screwdriver. If the engine continues to run rough with the kill wire removed, then the only spark related issue can be timing related.
Timing related spark issues would be (1) sheared flywheel key; (2) Coil upside down; (3) coil gap to magnet on flywheel; (4) wrong or bad parts (coil, flywheel, plug, plug wire); (5) if your insulation on the plug wire is weak, cracked or chewed by mice, the spark signal can be messed up as it will jump the air gap to the engine block (this will cause missing);
I suggest, going through systematically to verify each (spark, fuel, air, compression, valve lash; etc.) one system at a time.
As for running lean; Check air flow; make sure you don't have any air drawing in the system between carburetor and engine; (i.e. check the gasket between the carburetor and the engine; make sure the bolts are tight). You can also spray carb cleaner; or similar at the gasket point while the engine is running (make sure it doesn't enter the front of the carburetor though)... If it starts to run smoothly when you squirt it with carb cleaner (and it is not going into the front of the carb), then you have an air leak between the carb and engine. Also check the throttle and choke shafts where they enter the carburetor. If these are worn, you can get air leaking in there. Also check to make sure your throttle/choke plates are securely attached to the shafts (by looking into the input side of the carburetor)
Another test for running lean, is to move the choke to slowly towards the closed position and see if it starts to run smoothly.
Others may be able to chime in similar tests, but I would suggest going through systematically and prove out each system.
