“Turn off” and “turn out” are both perfectly ok; one phrase or the other might be preferred in some places by some people. Bear in mind though that they are only interchangeable when talking about certain things: the lights, for example, as you have said.
Yes in this context they both work however it is about the only situation where this can happen.
In general, "Turn off" means to shut down or switch off like is expected. "turn out" is usually used to say "the turn out was good" meaning a lot of people came.
Hope that helps
Yes, you are right. They have similar meaning in this context. Also note that you wrote turns X out / turns X off. You can also say turns out X / turns off X such as:
"My little brother doesn't go to sleep when Dad turns out/off the light(s)."
No one mentioned other synonym which I will list:
--switch on/off (the light(s)).
--turn on/off (the light(s)).
--push on/off (the light(s)). This example would be acceptable if the light(s) has a remote control (or a push-button). I've seen some houses that have those to control the lights.
--clap on/off This example would apply if you use a device that controls your lights by clapping .