Learn Spanish (Latin America): Él es doctor y ella es estudiante.

You are right that pronouns are often dropped! Because the verb es is used for both "he is" and "she is," dropping the pronouns would leave us with Es doctor y es estudiante. Using él and ella clarifies exactly who we are talking about. It also creates a strong contrast: "He is a doctor, and she is a student."

Why does Él have an accent mark over the 'e'?

The accent mark is used to distinguish él (he) from el (the). Both words are pronounced exactly the same, so the accent is just there to help you tell them apart in writing.

Why does the Spanish sentence leave out the word for "a" before doctor and estudiante?

In Spanish, you do not use words like un or una (a/an) when stating someone's profession or occupation right after the verb ser (to be). You just say Él es doctor (literally "He is doctor").

Does the word estudiante change to end in "-o" or "-a" depending on the person?

No, estudiante is a gender-neutral noun that stays exactly the same whether you are talking about a man or a woman. Because the sentence says ella es estudiante, we already know we are talking about a female student.