Learn Swedish: Jag bromsade, så bilen stannade.

No, it doesn't. When så means "so" or "therefore" (showing a consequence), it acts as a coordinating conjunction. This means it just glues two independent sentences together. The second part keeps the normal order: subject (bilen) followed by the verb (stannade).

I thought stanna meant "to stay". Why does it mean "stopped" here?

The verb stanna is versatile and can mean both "to stay" and "to stop", depending entirely on the context. Since the first part of the sentence is about braking (bromsade), the car clearly "stopped" rather than "stayed".

Both verbs end in -ade. Is this the standard past tense ending in Swedish?

It is for Group 1 verbs! Both bromsa (to brake) and stanna (to stop) belong to Group 1, which is the largest verb category in Swedish. For these verbs, you form the past tense by simply keeping the -a at the end of the infinitive and adding -de.

Is the så in this sentence the same as the one in så bra (so good)?

It is the exact same word, but it plays a different role. In så bra, it acts as an adverb meaning "very" or "to that extent". Here, så acts as a conjunction connecting two clauses to show consequence, translating closer to "therefore" or "as a result".