Electro, good to hear you had a good trip and good news about the surviving mosfet.
Sounds like you have a good method for checking the isolation of your power supplies. You can also check by measuring the voltage difference between your 2 0v from your different power supplies when they are not connected together, you are likely to see they are floating around. Note: your multimeter might bring them together, i know its annoying
Sounds like your soldering method is not ideal but seems like the best you can do unless you look at using a different mosfet that has a tab with a hole so you can screw it on rather than solder it. At the end of the day if the mosfet works the soldering doesn't matter
Regarding your switching frequency of 16 8 and 4 they are all in the audible frequency range so you are likely to hear it when your circuit is working which you could think of as a feature
. If it gets too annoying you can look at increasing it to 30kHz or above.
Looks like you dont need that gate resistor, you should just take it out. the gate itself is isolated anyway so current isn't going to flow. If you change to something other than a mosfet then you would look at using a gate resistor again.
When i finished up with university, there was a whole lot i had no idea about. those first few months developing electronic products was a steep learning curve.
So looking at this oscillation, it is HUGE!. Due to the time base on your image which is 200ns id assume this is mostly ESR related. I am guessing you have several huge electrolytic caps leave them as they are because they will hold up the power during a pulse of your 16kHz but add some more caps physically close to the mosfet rails these caps dont need to be a really high value, the ESR is the important thing, they could be 100 uF maybe, i would have to do the sums on your oscillation frequency and current draw but if it was me i would just put 100uF across the rails really close to the mosfet and see what happens. A contributing factor to this oscillation may be due to inductance/resistance between the cap and the mosfets.
PS you are interested here in the driver caps not the controller caps. Hopefully you have enough capacitance on your controller card and nice stable voltage rails on the controller card otherwise your drive signal would be almost unusable. If your controller card doesnt have stable voltage rails seperate the voltage rails of the controller card and driver card and put a filter into the controller card (inductor and capacitor on the voltage rail)
As you have indicated your voltage swing will just get much worse if you drive higher than 14A so best to get this fixed before increasing current otherwise your might blow things up. This might explain why you blow transistors with your starter motor and not with your smaller motor.
I shall look forward to seeing what you find