Learn French: Le détective commence son enquête. - The detective is starting his investigation.

Yes, enquête is indeed feminine. However, because it starts with a vowel, French uses the masculine possessive son instead of the feminine sa to make it easier to pronounce. Saying 'sa enquête' would create an awkward vowel clash, so son enquête flows much better! This rule also applies to mon and ton.

The English translation says 'is starting', but commence just means 'starts'. Why isn't there a translation for the word 'is'?

French does not have a separate 'present continuous' tense like English does. The standard present tense in French covers both general facts ('he starts') and actions happening right now ('he is starting'). Because of this, commence perfectly handles both meanings without needing an extra verb.

How do we know if Le détective is a man or a woman?

In this specific sentence, we know the detective is male because of the masculine article Le. If the detective were a woman, the sentence would start with La détective. The noun itself is spelled exactly the same way for both genders.