Learn Spanish (Latin America): Él me responde.

In Spanish, indirect object pronouns like me (to me) go right in front of the conjugated verb. While in English we say 'He answers me', the Spanish structure acts like 'He to me answers'.

Is me an indirect or direct object pronoun here?

It is an indirect object pronoun. With communication verbs like responder (to answer or reply), the person receiving the message is the indirect object. It helps to think of this sentence as 'He replies to me'.

Why does Él have an accent mark?

The written accent mark tells us this word is the pronoun Él (He). Without the accent mark, el is the article meaning 'the' (as in el libro). They are pronounced exactly the same way, so the accent just prevents confusion in writing.

Can I just say Me responde without Él?

Yes, absolutely! Because the verb ending in responde already tells us the subject is he, she, or you (formal), you can drop Él if it is already clear from the conversation who you are talking about. Adding it just gives a little extra emphasis or clarity.