Learn Farsi: داور مداوم دویده بود. - The referee had run continuously.

This is exactly how Farsi handles adverbs of manner! As your current lesson shows, you can often use a bare adjective directly as an adverb without changing its form or adding a suffix. Because مداوم (continuous) is modifying the verb here, it naturally acts as an adverb and translates to 'continuously'.

Could I also say به طور مداوم instead of just مداوم here?

Yes, you could. Adding به طور (in the manner of) before an adjective is the more traditional way to create an adverb. However, using the bare adjective مداوم by itself is perfectly correct, much shorter, and very common in both everyday speech and writing.

Let's review the verb: why is it دویده بود instead of just دوید?

دویده بود is the past perfect tense, meaning 'had run'. It emphasizes that the action happened further back in the past. It is formed by taking the past participle دویده (run) and adding the simple past of the 'to be' verb for the third person, which is بود (was/had). Using دوید would just be the simple past, meaning 'ran'.

I've seen داور before. Does it only refer to sports referees?

While داور is the standard word for a sports referee or umpire, it broadly means an arbitrator, judge, or person who makes a ruling between parties. Keep in mind that for a formal legal judge in a courtroom, you would usually see the word قاضی instead.