No, do not connect the 12V to the Mega Vin or power plug. Powering through Vin or the power jack means that the Arduino and all peripherals that are on the 5V rail are powered by the onboard 5V regulator. The on board 5V regulator is not heat sinked so will supply limited current before it overheats and shuts down. I would use a buck converter to drop the 12V to 5V and connect that to the 5V on the Arduino, bypassing the, weak, 5V regulator. Then the rated current of the DC DC converter is available on the 5V line.
With the buck converter connected to the 5V pin the LEDs, encoder, HC05, Hall sensors, buzzer and LCD will be powered by the buck converter through the 5V pin. That is one reason that I suggested at least 1.5 times the current required by the motors for the 12V supply. In fact, if you buy a cheap Chinese power supply, I would go for 2 times the current. Too much current capability will not hurt anything. Not enough certainly will.