Learn Spanish (Latin America): El actor habla español correctamente.

To turn an adjective into an adverb (similar to adding -ly in English), you start with the feminine singular form of the adjective. The feminine form of correcto is correcta. Then, you just add -mente to the end, giving us correctamente.

Why isn't the word español capitalized in this sentence?

In Spanish, the names of languages and nationalities are never capitalized unless they appear at the very beginning of a sentence. So, it is always written as español in the middle of a sentence.

Could I change the word order and say "El actor habla correctamente español"?

It is much more natural to keep it as habla español correctamente. In Spanish, adverbs that describe how an action is done generally go at the end of the phrase, after the verb and the thing being acted upon (in this case, the language).

If the sentence was about an actress, would it be la actora?

No, the word for actor has a special, irregular feminine form. An actress is la actriz, so the sentence would change to "La actriz habla español correctamente".