1.I feel like watching TV. 2. I fell asleep at 9 30 pm. -- 'Fall asleep' is an action at a point in time; 'sleep' is durational: 'I slept from 9 PM to 5 AM.'
1-- Sleep is a verb or noun. Asleep is an predicate adjective or adverb. 2-- 'Sleep off' = sleep in order to remedy a condition: I slept off my hangover and awoke at noon feeling OK. You are confusing 'drop off to sleep', I think. That means 'fall asleep'.