Learn French: Elle arrive du théâtre. - She is arriving from the theater.

du means "from the". It is a mandatory contraction of the preposition de (from) and the masculine word for "the", which is le. In French, de and le always squish together to make du. You will never see de le théâtre.

Would this change if she was arriving from a feminine place, like the train station (la gare)?

Yes! The contraction du only happens with masculine words. For a feminine place like la gare, you just use de la side-by-side without combining them: Elle arrive de la gare.

The translation says "is arriving", but doesn't arrive just mean "arrives"?

It means both! French uses the exact same present tense for facts (she arrives every day) and actions happening right now (she is arriving in this moment). There is no separate "-ing" tense in French.

How can I tell from this sentence that théâtre is a masculine word?

The word du gives it away! Because du is always the combination of de + le (the masculine "the"), seeing it right before théâtre is a built-in clue that the building is a masculine noun.