On a pure 240V circuit, no. There is no neutral. On a 240/120V circuit, yes. You have the needed neutral. Tapping 120volts off an existing 240volt branch circuit with a neutral is possible but is not a proper method. It does not comply with most electrical codes.
You have three feed wires on US residential service: 2 hots and neutral. These wires are connected to a 240V center tapped transformer with the neutral connected to the center tap. So, you have 120V between either hot and neutral and 240V between the hots. There is no neutral in this circuit because the load doesn't require 120V, it draws 240V directly from both hots.
You have to replace the wire (as you are increasing the current capacity), the outlet, and the breaker. Essentially you have to remove the old circuit and put in a new one. You can't reuse parts of the old circuit as you are increasing the current capacity and they would be underrated.
You DON'T! The washer needs to be on a separate dedicated circuit; 120v, 20 amps. The dryer needs to be on a 240 V, 30 amp dedicated circuit. Trying to do otherwise is unsafe.
120V takes less energy
In America, a 2-pole breaker is controlling 240V. 120V per leg.
Definitely not - it would burn out quickly and probably cause the circuit breaker to trip as well.
I don't know if you are trying to ask if you can run a 240V panel off a GFCI or run a 120 V sub panel off of a GFCI. Can you clarify please,,,Thanks
Yes it will operate it fine.