Sounds like you bought an abused motor. And it may have more than one problem. After 5 years of use, almost anything could have happened in the motor's history.
Only a severe overheat would melt the filler cap. You may have a bad thermostat or salt/crud blocking some passages in the powerhead. You can check the cooling system temperatures with a cheap IR gun-style thermometer. A properly-operating stat should maintain temps around 150/160 or so (they vary depending on the area of the block/head), as long as the cooling system is clear. The pee tube only shows that the cooling system is pressurized, so it's a good confirmation that you are pumping, but does not confirm circulation.
All EPA-rated 4-stroke carbs of all brands and sizes require super-clean fuel. If you feed it crud, it will choke. The tank, filters, and lines must be impeccably clean. You might consider a 10-micron water-separating filter. Gas should be less than 30 days old, preferably without ethanol. A proper carb cleaning requires removal and complete disassembly of the carb, then a 4-hour bath in real carb dip... then a blow-out with generic carb spray, and a careful reassembly. If that was not done, the carb was not thoroughly cleaned. You must leave the carb empty at the end of the day. If you forget to run it out, you can open the drain screw and let the gas out. Failing to do that allows the gas to evaporate from the carb, leaving varnish deposits in the passages.
All 4-stroke outboards are sensitive to crankcase oil levels. Keep the level about 1/2 way on the dipstick -- never even a drop above full. Run only non-synthetic, NMMA-certified FC-W oil, either 10w-30 or if really hot, 10w-40. Sounds like oil is spilling into the lower cowl when cranking. You may be over-full, or a breather hose may be off, or the filler cap may be off, or you may have a gasket leak of some sort.
So, you have some diagnosis to do. If you want to proceed yourself, get the Factory service manual and review the cooling system, and the fuel system. It's also possible that you have a heat-triggered electrical issue. Once you have the cooling and fuel systems up to par, you can see if the motor still stalls at speed when hot.
In order to get the right performance from the rig, especially with the somewhat-heavier 4-stroke... prop pitch, jack height and trim angle are critical. That can only be adjusted by trial and error. Get the motor itself healthy first, then review the rig on the transom. Try to keep batteries and fuel tanks forward, for better weight distribution.