I have never heard of turn out the lights. The correct way is turn off the lights. I guess it could depend on where you live in the U.S like Miss Marianne said ^ . Here in the south of the U.S you will hear " turn off the light" instead of "turn out the lights".

Yes, both are acceptable.

Yes. It often depends on what part of the U.S. you live in. I use both!

turn off