Drain the Fuel and Carb bowl. Check emulsion Tube holes if you can access them. There is often a very tiny hole on the jet at the bottom of the float bowl. A wire from a wire brush will handle that. Carefully remove Float and needle valve and allow fresh clean gas to wash through the seat...
You can replace the carb or just unplug the Solenoid from the wiring harness.
Install a good Shut Off valve on the fuel line and use it every time.
Shut it off about 20 seconds before you are done with the mower.
Grey oil indicates metal, likely from the Connecting Rod Big end. Use Standard 30 oil. Likely a lot of oil inside the muffler causing smoke, so contamination (Aluminum Alloy) may be coming from the Cylinder Wall.
Don't over think it. Keep your eye out for a good newer model. Tis the season!
The only way fuel can get into the Crankcase is from the Carb. A spec of dirt at the needle/seat interface is all it takes.
The fuel Shutoff valve may not be getting power or is defective. You should be able to feel/hear it click when slowly turning the key ON.
It is usually possible to...
First, Remove the blades and sharpen them. Not rocket science as some might say. Then try it. No sense replacing belts and idlers yet as it seems your mechanical skill level in this is quite low.
If your mower is at operating speed in the photo, it appears the Throttle Plate is nearly closed.
Governor linkage can do that if there is a fuel problem (not enough gas). Try squirting a small amount of gasoline into the Carb when it is set for wide open and see if it revs up considerably. I...
If you hold your finger on that solenoid you should feel it click when you slowly turn the key towards but not to Start.
Warm the fitting briefly with a propane torch and then cold water. Do that a couple times then with a tight fitting wrench try to tighten then loosen the fitting. You could...
I expect you have checked for/changed the fuel filter. Remove Float bowl and see if fuel runs out when cranking engine.
Remove Float and needle and use carb spray to spray back into the NEEDLE SEAT.
If you fill several 5 gal cans of fuel at the lowest price you can find and they are tightly sealed, no problem.
It is the moisture in the Air in partially filled tanks that offers the Ethanol moisture.
The fuel will normally last 30 days with no degradation.
I use Premium Ethanol Free in...
I believe you have a direct short somewhere . A fuse would not allow that type of damage if it was in that circuit.
High resistance somewhere or melted insulation. Follow each wire, replace any overheated connedtions and clean all mating surfaces that wires connect to. Good Luck.