Learn Italian: Anche se sono disidratato, non mi arrendo. - Even though I am dehydrated, I do not give up.

Anche se takes the indicative mood when you are stating a confirmed fact. Here, the speaker is stating that they are actually dehydrated. If they had used the subjunctive (Anche se fossi disidratato), it would change the meaning to a hypothetical: "Even if I were dehydrated."

Why is there a mi before arrendo?

The verb for "to give up" or "to surrender" is arrendersi, which is reflexive. Since the subject is "I", you must use the reflexive pronoun mi to make mi arrendo.

Would a female speaker say this sentence exactly the same way?

No, because disidratato acts as an adjective describing the subject "I" (who is male in this sentence). A woman would say Anche se sono disidratata. The rest of the sentence, including mi arrendo, remains exactly the same.

Could I use a gerund here to say "Despite being dehydrated"?

Yes, but not with anche se. Italian does not put a gerund directly after anche se. To use a gerund, you would swap anche se for pur: Pur essendo disidratato, non mi arrendo.