Lawnmower engines are reliable devices, and should not require any adjustment. The main thing that should be adjusted or replaced is the throttle cable length and position, and the governor springs.
It is a carburetor where the engine coolant is routed through the choke system of the carburetor. When the coolant is cold the choke will close. As the engine coolant heats up it opens the choke.
Check your engine timing, it may be too slow.
Chainsaw carbs are very picky and have to be set just right. As you are obviously not familiar with them, you should take it to you local lawnmower and small engine shop.
Mounted on the side of the engine if it has one. Follow the fuel line as it goes to the pump. If the fuel line goes to the carburetor then it does not have a pump and is gravity fed.
Most common problem is a dirty/clogged fuel filter restricting a good flow of fuel to the engine. Otherwise, the carburetor needs adjustment. Or, if the engine is blowing black smoke before it dies, check the spark plug and the air filter, or a too rich mixture from the carburetor.
The coolant should circulate through the entire water system.
More than likely this means your carburetor float has stuck at some point and let gas leak into the engine after it was shut off. To be safe you really need to replace the carburetor or have it cleaned.
A choke on a lawnmower engine, or any engine with a carburetor, is a device that reduces the airflow through the carburetor. It causes the fuel-air ratio to be richer, that is, more fuel and less air. This helps most carbureted engines to start while cold and before steady fuel flow is established, but the rich mixture reduces the power and efficiency of the engine tremendously. Leaving the choke on will produce hydrocarbon and carbon monoxide emissions and may cause the engine to stall as soon as any load is placed upon it. That is why engines require either manual release of the choke or have an automated device to back it off. Excessive use of the choke will generally result in fouled spark plugs.
Either through a carburetor or by a fuel injector.
Incorrect timing or plug wires connected incorrectly.
Yes it has a carburetor. An engine with fuel injection would be a fuel injected engine.