Learn Finnish: Hyvää huomenta, opettaja!

They are in the partitive case. When you wish someone well in Finnish, like saying "Good morning" or "Good night", you use this case. It is as if you are saying "I wish you some good morning." So hyvä becomes hyvää, and the old word for morning (huomen) becomes huomenta.

I thought the word for morning was aamu. Why does this greeting use huomenta?

Huomen is an old, traditional Finnish word for morning. Today, it is almost exclusively used in this specific greeting (Hyvää huomenta) and as the root for the word tomorrow (huomenna). When talking about mornings in normal sentences, you will use the modern word aamu.

Why doesn't opettaja also get a partitive ending like the rest of the greeting?

Because opettaja is the person you are speaking directly to, not the thing you are wishing them. In Finnish, when you address someone by their name or title to get their attention, you just use the basic dictionary form.

Why does hyvää end with -ä while huomenta ends with -a?