1. I fell asleep at 11:30 pm. Is it natural and correct? Does this mean I went to sleeping state at 11 30 pm.?-- Yes and yes.
2. I started sleeping at 11 30 pm.-- Not natural: 'sleep' is not a conscious action
3. I slept off at 11 30 pm.-- Not possible. 'Sleep off' is a transitive idiom meaning to sleep until the effect (of alcohol, drugs, melancholy.