Can I write my reply in my own language and send it in the customer's language?

Yes. When a ticket is translated, a button appears above the reply editor reading "Translate to {language}"; click it to translate your draft into the customer's language, and "Translate back to {language}" to switch it back. If you send a reply on a translated ticket, Deskhero asks "Do you want to send the reply in {target}? Original ticket language is {source}" so you can confirm the language before it is emailed.