Learn German: Der Zeuge musste vor Gericht schwören. - The witness had to swear in court.

In German, the standard idiom for "in court" is vor Gericht (literally "before court"). Notice that it does not use an article here, similar to the English phrase "in court." If you specifically meant being physically inside the courthouse building, you would say im Gerichtsgebäude.

Can schwören also mean "to use bad words" like it does in English?

No, schwören strictly means to take an oath or make a solemn promise. If you want to talk about swearing in the sense of using foul language or cursing, you need to use the verb fluchen.

Zeuge ends in an "-e". Why is it masculine?

While many German nouns ending in "-e" are feminine, der Zeuge (the witness) is a masculine noun belonging to the "n-declension" group. This means it takes an -n ending in all cases except the nominative singular. It is the subject here, so it stays der Zeuge, but as a direct object it would be den Zeugen.

Why use the simple past musste instead of the conversational past?

Even though spoken German normally prefers the conversational past (Perfekt) for most verbs, modal verbs like müssen are a major exception. Native speakers almost always use the simple past (Präteritum) forms like musste, konnte, or wollte for modals, even in very casual speech, because the Perfekt version is clunky.