Learn French: Désolé Madame, le restaurant ferme maintenant. - Sorry madam, the restaurant is closing now.

French does not have an "-ing" verb form like English does. The present tense verb ferme can mean either "closes" or "is closing" depending on the context. Since the action is happening maintenant (now), "is closing" is the most natural translation.

Would a female restaurant worker also say Désolé?

In spoken French, yes, it sounds exactly the same. But in writing, a woman would add an extra "e" to make it Désolée. This is because désolé is an adjective describing the speaker, so its spelling must match the speaker's gender.

Why isn't there a word for "I am" before Désolé?

Just like in English where we can shorten "I am sorry" to just "Sorry", you can drop Je suis (I am) in French and simply say Désolé in everyday conversation.

Do I need to say "la" before Madame?

No, when you are talking directly to someone, you just use Madame (or Monsieur) on its own without a word for "the". You would only use la (like in la dame) when you are talking about the lady to someone else.