There's a lot going and bear in mind that I'm only self-taught in electronics.
Your FETs stuck on because that's usually what happens when they fail. The gate doesn't need to have a diode on it(see 2nd paragraph); it only needs a discharge resistor to ground so it will shut off quickly. That's the thing with FETs; they have to be switched very quickly so they're either fully on or fully off. The more time they spend in the transition, the more heat they dissipate. The gate resistor value vs. switching time can be calculated using the datasheet's gate capacitance. 1 megaohm is way too high for this application. It's likely going to be in the hundreds of ohms, not millions, but also keep in mind that the resistor goes directly to ground and current through it when the gate is on should not exceed the resistor's rating. There should also be a resistor between the gate and whatever is supplying the gate signal(the 555?). The gate capacitance will pull a lot of initial current when it starts charging and that could exceed the signal source's rating, eventually damaging it.
Flyback diodes are reverse biased on the load. When the FET shuts off an inductive load like a solenoid or motor winding, the field collapse creates a high reverse voltage spike (the flyback voltage) on the FET's drain/source pins that exceeds the FET's breakdown voltage and will damage it. A flyback diode acts as a backward short for that voltage and prevents it from damaging the FET. Shottky barrier diodes are preferred for this role because they have very fast switching characteristics.
Your FETS are also way too small for the job. When you look at the datasheet's "Absolute Maximum Ratings" section and it shows 50A of current or something, there's almost no practical way to keep it cool at that current and the current capacity goes down rapidly with temperature(shown in a chart in the datasheet). You would need probably 10 or more of those FETs in parallel with good cooling to handle a 100A load reliably.
That brings me to the final point: your motor probably draws more than 100A at low speed and high load. It would be helpful to know what the original mower's advertised specs were(wattage, horsepower, whatever). Barring that, you can try to measure the winding resistance and deduce how much current they're capable of pulling at 36V.
