Caught
I'm going to give 2 examples of the word "caught" with different contexts and tell me if I used it correctly or if you think that I misunderstood the idea of this word because I'm kind of confused how to apply this word in a sentence

Example 1
I caught him sleeping when I arrived.
What I'm trying to say here is that he's in the act of sleeping when I arrived
Example 2
I caught him smoking
In this example I'm trying to point out that he's smoking and it's forbidden for him to smoke.
Example 3
I caught the bus just before it left.
In this example I'm trying to say that I'm in a rush and I reached that bus just before it left

If there are other ways to use this word in a different context please tell me and also tell me if there are more natural ways to express a sentence on the examples I've given. Thanks a lot

Question

Caught
I'm going to give 2 examples of the word "caught" with different contexts and tell me if I used it correctly or if you think that I misunderstood the idea of this word because I'm kind of confused how to apply this word in a sentence

Example 1
I caught him sleeping when I arrived. 
What I'm  trying to say here is that he's in the act of sleeping when I arrived 
Example 2
I caught him smoking 
In this example I'm trying to point out that he's smoking and it's forbidden for him to smoke. 
Example 3
I caught the bus just before it left. 
In this example I'm trying to say that I'm in a rush and I reached that bus just before it left

If there are other ways to use this word in a different context please tell me and also tell me if there are more natural ways to express a sentence on the examples I've given. Thanks a lot

Answer

Your examples are all pretty good.
'I caught him smoking' would also have that 'He was in the act of smoking when I arrived' idea, as well as the 'naughty' aspect.
That is probably the best word for those situations. Maybe 'just caught the bus' would be a better order to reinforce the urgency.
In terms of other uses of 'caught' there is also the obvious - 'I caught the ball'. Also 'I caught a cold', and 'I caught on as soon as she mentioned her father'. 'caught out' is a phrase meaning needing something like a toilet, when there isn't one around.