is there a point on the carrier board where I can draw 5V to power an Arduino Nano (19mA) - even if the board is not powered on? (The GPIO pins only turn on when the board is powered on.)
Otherwise, would it be safe to hook up to the 20V jack and use an LDO?
No such point on board.
As for the LDO to 20V jack, it depends on the input voltage range request of LDO. Would be safe if in the range.