In the case of Takahero's comedian perhaps not, although I'm not so sure. But it can be in other cases. I had the mechanic mend the car means that the job was finished. The comedian had us laughing our heads off doesn't. It's a description. It takes us into the middle of the action.The difference is not one of completed actions vs. uncompleted ones.
Thank you.(Actually I had the children mowing the lawn could mean – although I agree it’s extremely unlikely – I gave birth to the children while I was mowing the lawn. But of course, Thomas doesn’t mean that.)
You seem to understand the use of the infinitive, Takahero: I had the waitress bring me a cup of tea means I asked the waitress to bring me a cup of tea and she brought it.
The meaning of have with the –ing form is the same. It’s just that there isn’t the same idea of completed action. The boss had me clean the windows – I cleaned them. The boss had me cleaning the windows – there’s no implication in this sentence that I finished cleaning them.
(Taked4700’s idea of the superiority of one person over another is an interesting one. Clearly the boss is superior to me in that he can tell me what to do. But the idea needn’t be strong. It’s nice to have people make a fuss of me. People can stop making a fuss of me at any time and I couldn’t do anything about it. But at least they’re making a fuss of me for my benefit.
In terms of register, the construction can convey the idea of officiousness. I had the waitress bring me a cup of tea almost sounds as if I was ordering her about. I'd be more likely to say I asked the waitress to bring me a cup of tea.)