How and when does a regulator draw energy?
Yes, regulators consume current even if nothing is connected to output. It requires current to operate and would not operate without current.

Yes, due to non-zero current being consumed, it would drain batteries faster than when batteries are disconnected.

Yes, current draw from regulator output is exactly the amount of current drawn by loads connected. The same current is drawn from batteries, plus the current required by the regulator itself to work.

Yes, regulator output current would increase when connecting loads. Current needed by the regulator can vary slightly, but not so much it would matter much in general. In your case it might. Note that two LEDs in parallel on a 12V regulator output would get damaged unless the LEDs are connected with a resistor in series.

Yes, power drawn from regulator is exactly the amount of power consumed by devices it powers. This does not include the power wasted in the regulator, as regulator input current and output current are approximately equal (if disregarding the current consumption of the regulator itself), but regulator input voltage is higher than output voltage. For instance if you have 24V battery, a 12V regulator, and 0.1A load, the load consumes 1.2W, but battery has to provide 2.4W, and the regulator converts the excess 1.2W to heat. Which is why linear regulators may not be suitable for every purpose.