“have someone V/Ving” in American English

Both patterns are possible in AmE, though they mean slightly different things and therefore aren’t completely exchangeable. Specifically, it mirrors the difference between the past and the past continuous tenses (or present and present continuous, etc., depending on what basic tense have has in your sentence).

It then becomes a question of whether the past or the past continuous is appropriate (or both).

• The bouncers threw them out (this is fine)
• The bouncers were throwing them out (there isn’t any context here indicating the progressive is appropriate, though there could be)

And the other:

• I did all kinds of jobs (fine)
• I was doing all kinds of jobs (also fine and puts a very slight emphasis on the fact that doing all those jobs took a while)

Here are some real examples:

• And then the next day, she called something else in because she felt it wasn't good enough, and they had the cops come and take my son away. (Talk of the Nation, transcript via COCA)

• She remains a spectacular talent, vivid in her interpretation of a song, not only vocally but in stage presence and in athletically powerful body language that had the audience cheering its lungs out. (Washington Post)

• Pulled the guy off... and then a shorter one came in from the small office... next to the kitchen where he had the bartender opening the safe. (NYPD Blue, transcript via COCA)

For the first example, it’s not possible to use the continuous. In the second, the continuous just sounds better to me because it’s used for a long, drawn out action, though the other form is also acceptable. (Though the meaning of had here is slightly different, the same grammar applies.) In the third example, the continuous is required because opening the safe was happening at the same time as everything else, in the background.