Learn Farsi: آمریکایی است و بیمار است.

In Farsi, you don't always need to say words like "I", "you", or "he/she" (the subject pronouns). The verb است (is) already tells us we are talking about a third person (he, she, or it). Because of this, leaving out the word for "he" (او) is very common and actually sounds more natural in everyday speech.

Could this sentence also mean "She is American and she is sick"?

Yes! Farsi does not have different words for "he" and "she". Even if the pronoun او (he/she) were used at the start of the sentence, it would still be completely gender-neutral. You usually figure out if the person is a "he" or "she" from the context of the conversation.

Why is است (is) used twice here? Could we just say it once at the end?

You can actually drop the first است and just say آمریکایی و بیمار است. However, keeping both verbs is also perfectly correct and is often done to emphasize both facts equally. Notice how است comes at the end of its phrase each time, because Farsi verbs almost always go at the very end of a clause!

What makes the word آمریکایی mean "American" instead of just "America"?

The base word is آمریکا (America). By adding the یی suffix at the end, it turns the country name into an adjective describing nationality, changing the meaning to "someone from America" or "American".