Learn Hardcore Japanese: kousaten de kuruma ga kyuu ni tomarimasita. - A car stopped suddenly at the intersection.

に would indicate a point of arrival or existence (“at/in”), not the scene of an action.
を marks a direct object, but 交差点 is not an object of stopping—it’s the place of the action.

Why is the car marked with が rather than を or は?

が marks the subject of an intransitive verb. 止まる is intransitive (“to stop by itself”), so the thing that stops takes が.
If you used を, you’d need the transitive verb 止める (“to stop something”).
は would make 車 the topic, which you could do for contrast or emphasis, but here が simply introduces new information (the car as the subject).

What is 急に, and why does 急 take に?

急 (きゅう) by itself is a noun or na-adjective stem meaning “urgent” or “sudden.” Adding に turns it into an adverb (急に) meaning “suddenly.” Adverbs often end in -に when derived from nouns or na-adjectives.

Could you say 急止まりました or omit に?

No. 急止まりました is ungrammatical. To modify the verb with an adverb you need 急に (or another adverbial form like 突然 or 急速に).

Why is the verb in the polite past form 止まりました? Could you use the plain form?

止まりました is the polite past tense, appropriate for formal speech or talking to someone you don’t know well.
In casual contexts among friends you could say 止まった instead.

If I wanted to say “The driver stopped the car” instead, how would I change the sentence?

You would switch to the transitive verb 止める, make 車 the direct object with を, and make the driver the subject. For example:
運転手が交差点で車を急に止めました。

Is it acceptable to change the word order, for example 車が交差点で急に止まりました?

Yes. Japanese is relatively flexible:

交差点で車が急に止まりました。 (sets the scene first)
車が交差点で急に止まりました。 (focuses first on the car)
Both are correct; choose based on what you want to emphasize.