Learn Portuguese (Portugal): Quando estou doente, fico em casa.

In Portuguese, there are two verbs for "to be": ser and estar. We use estar (here as estou) for temporary states. Being sick (doente) is a temporary condition. If you used sou, it would sound like you are a permanently sick person!

Why isn't there an eu (I) before the verbs?

You don't need to say eu estou or eu fico because the verb endings already tell us who is doing the action. Both estou and fico are the "I" forms of their verbs. Dropping the pronoun is very common and sounds more natural.

Does the word doente change depending on whether a man or a woman is speaking?

No, it stays exactly the same. While many Portuguese adjectives end in "-o" for masculine and "-a" for feminine, adjectives that end in "-e" like doente are usually gender-neutral. Both men and women say estou doente.

What exactly does fico mean here?

The verb ficar is extremely common in Portuguese! Its basic meaning is "to stay" or "to remain". In this sentence, fico simply means "I stay" in a location.