So, this use of "had" can have two meanings. It can mean, as you note, that you caused something to happen. It can also be a way of expressing something happening to you in the past, which is more a sense of possession.
In the first sense, "I had a bullet go through my arm" means that you caused a bullet to go through your arm. In the second sense, it means the same as "a bullet went through my arm".
If you have the progressive participle (not the gerund - they look the same, but they are different things) there, though, the progressive shifts the meaning. "I had a bullet going through my arm" means at the specific point in time, a bullet was passing through the arm. Some other objects might mean different; "I had an arrow going through my arm" might not mean the arrow is in motion, but that it is lodged in the arm with some of both ends sticking out. "I had a spike sticking out of" some body part would always have that static meaning, because that sense of stick isn't suggestive of movement or progression.
Sometimes, the progressive doesn't work for the 'cause' meaning. I think that's usually when the period of time the progressive would cover would be very short, like "I had a bullet going through my arm". "I had security throwing him out" can work, though, in the right context, in reference to a point in time where security were throwing him out on the speaker's orders.
So, by and large, in both senses, it can be either progressive or infinitive. It will have different meanings, and may or may not make sense depending on exactly what words are involved, and also depending on context.