You have several possibilities.
Just read all of my answer before you act.
1. Todays gasoline is not the same as it was just a few years ago. There have boon changes in gas formulas. DO NOT USE REGULAR 87 OCTANE GAS! Use a good quality, good brand 89 octane fuel. DO NOT USE CHEAP GAS.
2. One of the problems you seem to have is a phonomenon called "VAPOR Lock".
This is caused when fuel vaporizes in the fuel line and doesn't allow the proper amount of gas to get to the carburetor. This is usually caused bu the fuel ine getting too close to a heat source such as the muffler of the cylinder head or even sometimes the engine block. Also cheap fule of 87 octane fuels will cause this to happen more often. Check all the fuel lines and make sure that they are away from heat.
As mowers are operated, they tend to accumulate grass under the engine shoud. We call this a "Birds nest or mouse nest." These nests will keep the engine from cooling as they block the flow of air to the head and cylinder. Clean this grass out several times a year.
3. Another issue can be a fuel issue in the carburetor.
As engines sit or get older, fuel that is left in the carburetor can turn to gum and varnish and cause this and other problems.
Also, any gasoline that was left in a gas can for a period of more than 30 days must be discarded because it also has begun to turn to varnish.
More than 70% of all of our repairs in our lawn mower business are due to this same issue. You most likely have dirt, gum, varnish...etc in your carburetor plugging up the small passageways and jets in the carb. It Must be removed from the engine, cleaned very well, blown out with compressed air and reassembled using a NEW carburetor rebuild kit. ALWAYS clean the fuel tank and replace the fuel line when doing this repair or you may have to do it all over again. The inside of the fuel line disintegrates over time and these small pieces of rubber will plug up the carb too. Dirt and water from a dirty fuel tank will also plug up the carb. Find the Model, type and serial or code numbers off of the engine and take them to your local dealer to get the carb kit.
4. Make sure that you install a new Air Filter. DO NOT CLEAN AIR FILTERS always REPLACE. Replace air filters no less often than every 20 hrs of use. Plugged air filters will cause many problems.
Forgive some repetition below, but it is needed to maintain the flow of the answer..
This problem is common enough that I have put a template together to paste in and modify just in case.
What could cause the overheating problem?
Overheating can be attributed to two things something getting to hot or something not getting cooled properly.
Lets take the situations that would cause something to get too hot first.
Engines can create extra heat by having the engine run too lean....(Not getting enough fuel) Fuel is not only used to create the power in an engine but it is also used as a coolant in the combustion chamber. If it runs too lean, then the explosions will create more heat and then there is also not enough fuel to help cool the engine.
What would cause the engine to run too lean?
Ok now lets see what could cause the engine to have insufficient cooling.
When small engines run, they build up heat as a natural byproduct of combustion.
Just as a wood burning stove gives off heat so does a combustion engine. Different fuels will produce different amounts of heat. Coal for instance burns much hotter than wood.....a hardwood such as oak or hickory burns much hotter than softwoods such as soft maple of cottonwood. The same can be said for gasoline. 87 octane gives off much more heat than 89 octane of premium. That is because 87 octane explodes and all of the heat is transferred to the piston in one very fast burst. It cannot be transferred through the piston and cylinder to the air fins where the air from the cooling fan takes it away fast enough to keep the engine running as cool as we would like. On the other hand, mid-grad 89 octane and premium fuels do not explode...they burn in a much slower manner and the added time aids the engine in cooling., so the engine runs much cooler. This is due to the antiknock compounds that have been added to the fuel. The tiit-knock compounds job is to slow the burning process and to eliminate the explosions that result in the knock (or pre-ignition) that occurs with regular gas. The problems we are having with the gas these days is that they keep making it cheaper and cheaper by not adding as many additives as they one did. These additives are very expensive.
So as a conclusion, we do not recommend using 87 octane in ANY SMALL ENGINE. We would prefer that only good Brand (NOT CHEAP DISCOUNT STATION) gasoline is used in all 4-cycle engines and Premium MUST be used in ALL 2-cycle engines.
Another situation that causes the engine not to cool well is that grass builds up under the engine cover....we call this the birds nest affect.....When grass is built up under the shroud or cover as we call it, it blocks the flow of air around the cylinder and block of the engine. This prevent the engine from cooling because it cannot take away the heat that was created during combustion. In many cases this grass build-up gets so hot that it can even create a fire under the cover.
What can be causing the engine to be hard to start or in your case now, impossible to start.
When the electronic ignitions get hot, they have a tendency to short out on the inside. Sometimes they will run ok for a while 10 min, 20 min, an hour etc before they get hot enough to break down under the heat. Then they will short out and not give any spakt to the engine and it won't run. IT will die. Sometimes if you let the engine cool of for an hour or two, the ignition will begin firing again...and the cycle starts all over again. At some point, the ignition gets hot enough that the short inside doesn't go away when it cools off. And it goes permanently dead. Sometimes the ignition goes dead with no other warning. You simply have to replace the electronic ignition coil for the unit to run again.
There are other causes that will keep an engine from starting too, such as the valves may need to be adjusted or replaced...sometimes because of overheating and sometimes just because of normal wear and tear. If the valves are out of adjustment the engine will not run or at least will not run correctly. Also.. if the engine has been overheating, the valves could be burnt and then they will not seat correctly.
I hope that these suggestion will help you in you're quest to find the problem with your engine. If I can be any more help to you , please just ask for me.
There may be other things, but this should be enough for now.
There is not really an easy way to reliably test the coil.
If you test the coil and it tests bad, then it is bad..
If you test the coil and it tests good, it could still be bad..
It sounds like it is bad because it is breaking down with heat..
At the shop we have the benefit of being able to grab one and throw it on..
I suppose you could run around with a spark tester in your pocket..
Run it til it quits.
Then JUMP off and quickly test the spak..
No spark means bad coil in this case.