dhenry:
When I use a 12V, 2A, center positive power source it resets itself every once in a while and gets very hot.
In that set-up, if you need more than 200ma, I would seriously question if something is wrong.
Not sure what point you're trying to make.
Ethernet shields take 150mA or more, and the LCD probably needs 75mA or more, and Arduino 40mA, all adds up to more than 200mA - hence the excessive heat from 12V leading to thermal-shutdown of the on-board regulator.
Solution is either good USB PSU, or 7..9V supply to the barrel jack, or even a DC-DC converter (good choice for vehicle as supply is 12V anyway)