Learn French: Le professeur est indulgent, mais le client ne l'est pas. - The teacher is lenient, but the client is not.

It replaces the adjective indulgent. In English, we can just end the sentence with the verb: "the client is not." French doesn't allow this structure. You must use the neuter pronoun le (which elides to l' here) to represent the previously mentioned adjective or state.

If the sentence were about a female client, would we use la instead of le?

No, it would still be l' (representing le). When this pronoun is used as a predicate attribute to replace an adjective, it is invariable. It never changes to match the gender or number of the noun, so it is always le (or l' before a vowel) even for feminine or plural subjects.

Why is the l' placed inside the negative ne ... pas?

Because le is an object pronoun, and French object pronouns always go directly before the conjugated verb. The negative structure acts like a sandwich that must wrap around the entire pronoun-verb group together, resulting in ne l'est pas.

Does indulgent mean the same thing in French as in English?

It is a partial false friend. In French, indulgent specifically means lenient, tolerant, or forgiving of mistakes. It isn't used to describe treating yourself to luxuries or rich food.