Learn Arabic (Modern Standard): عزيزي أحمد، تحية طيبة.

In written correspondence like letters or emails, it is very common to drop the vocative particle يا. You simply start directly with عزيزي (My dear) followed by the person's name.

Is تحية طيبة considered a complete sentence here?

No, it translates literally to just a good greeting. Both words use the feminine marker ـة so they match. This is a fixed formula used to open messages, similar to writing Greetings in English before starting the main body of text.

If I am writing to a woman, does the word عزيزي stay the same?

No, عزيزي is used here because Ahmad is male. If you were writing to a woman, you would use the feminine base word عزيزة before adding the my suffix, which makes it عزيزتي.

What usually follows this greeting in a formal Arabic letter?

Almost always, you will see this followed by the word وبعد (and after). The full phrase تحية طيبة وبعد is the classic way to transition from the polite greeting to the actual subject of your message.