What are the two big transistors on a ESC

I suspect the resistance is the resistance of the internal power MOSFET(s). In which case it's an utterly pointless measurement as the internal resistance of the battery, the resistance of copper tracks, and the virtual "resistance" created by charging and discharging the MOSFET's gate will easily overwhelm this tiny Rds(on).

If you want to know the effective resistance of your ESC you can perform an experiment. Set up a motor on your ESC. Wire it in for maximum speed (this would probably involve wiring the throttle input to a battery line.) Turn on the motor, making sure nothing is attached to the rotor. Put an ammeter in series - preferably a clamp-on one - current will easily exceed the 10A rating of most series meters - and measure the current going to the motor.

Then, measure the voltage across the battery, and across the motor. You could try measuring the drop across the ESC as well but it wouldn't work if it employs some virtual ground or has a current shunt built in, as this will skew your readings. Calculate the voltage drop across the ESC and put this into Ohm's law: R = V/I. Note this is only an average resistance and depends on the state of the battery.

For yuks, measure the current from the battery to the ESC and use that to make input-output power figures. The efficiency is the output divided by the input, in percent. This will probably be more useful than any resistance figure.

It doesn't, the motor itself does. Once the rotor starts spinning, the motor produces a voltage that opposes the flow of current; this is commonly called "back EMF (electromotive force)".

The motor's speed increases until the back EMF reduces the current flow to the level needed to account for the actual physical load on the motor (plus losses).

The heavy current you calculate is drawn only for an instant, just as the rotor starts spinning. If the rotor is prevented from spinning, then that current will be drawn indefinitely, and yes, it can destroy the motor.