Questions & Answers about Ele é um médico muito bom.
I learned that we don't use um or uma before professions in Portuguese. Why is um used here?
You're right that we normally drop the article for professions (for example, ele é médico simply means "he is a doctor"). However, when you add a description to the profession—like muito bom (very good) in this sentence—you must use the indefinite article (um or uma).
If we were talking about a female doctor, how would the sentence change?
Why is the order médico muito bom instead of muito bom médico?
In English, you put the adjectives before the noun ("a very good doctor"). In Portuguese, adjectives almost always come after the noun they describe. So, you say the equivalent of "a doctor very good" (um médico muito bom).
Why does this sentence use é and not está?