prevented or had prevented

I'd say the simple past ("prevented") is better than than past perfect ("had prevented") because the sentence is describing two events in the past ("she made a sign" and "police officers [prevented/had prevented] her from doing it") and the second one clearly happened after the first one. The past perfect would be used only if the second event happened before the first one, since that is the function of the past perfect (to describe an event that ended before a definable time). If anything, the first event could be put into the past perfect...

"She had made a sign trying to say something but the police officers prevented her from doing it."

...but the author has chosen not to do it, most likely because the sentence is trying to narrate the sequence of events in "real time" (i.e. to relate them as if they were unfolding, event-by-event, in the present). Either way, the simple past is preferable for the second event so I would go with "prevented."

Yes, and there's nothing wrong with that.

All those tenses are rather confusing to think about.

In terms of language think of time as a 1D line with zero at the current time. Anything before that is past (negative), anything after is positive.

However, when you are having a conversation you can move the origin forward or backwards.

"Prevented" is for talking about events that are actually at the "origin" when the origin has been moved into the past.

"Had prevented" is for talking about events before the origin.

You should also think about changing "in a violent way" to "violently." They are both correct but this is a more common way of writing it and therefore sounds more fluid to native speakers.

Yes, if the aim of the passage is simply to give an account of what happened, you're better off putting everything in the past tense (or "simple past"). The other forms of the past tense are useful if you want to give the account a more nuanced or more literary flair, but even then it's not necessary and can sometimes be tedious. In conversational English the simple tense is more than enough.