Learn German: Der unruhige Patient musste beruhigt werden. - The restless patient had to be calmed down.

Because of the modal verb musste. In a standard past passive sentence without a modal, you would use wurde or ist ... worden. However, when you add a modal verb, it takes the conjugated spot (musste), pushing the passive action to the end of the sentence. Modal verbs require an infinitive, so the passive auxiliary reverts to the infinitive werden, resulting in the structure: conjugated past modal + past participle + werden.

Could I express "had to be calmed down" in the spoken past (Perfekt) instead of the simple past (Präteritum) here?

You could, but German speakers rarely do. Modal verbs in the Perfekt tense use a "double infinitive" structure when combined with another verb. A passive sentence with a modal in the Perfekt would look like this: Der unruhige Patient hat beruhigt werden müssen. Because this is so clunky, Germans heavily prefer the Präteritum (musste beruhigt werden) for modals, even in casual spoken language.

How do the words unruhige and beruhigt relate to each other?

They both share the root noun die Ruhe (peace or quiet). Adding the suffix -ig creates the adjective ruhig (calm), and the prefix un- makes it unruhig (restless). To create the verb, German uses the prefix be- to make beruhigen (to calm someone down). In this sentence, we see its past participle, beruhigt.