Öğretmene günaydın dedim.

Because öğretmene has the dative case ending -e, which often means to or toward.

• öğretmen = teacher
• öğretmene = to the teacher

With demek in this kind of sentence, the person you say something to is usually marked with the dative case.

Here, -e marks the recipient of what is being said.

So in:

• Öğretmene günaydın dedim

the structure is basically:

• to the teacher
  • good morning
    • I said

This is very common in Turkish. Many verbs use case endings instead of separate words like to.

Turkish commonly uses demek for expressions such as greetings, short statements, and direct speech.

So:

• günaydın demek = to say good morning
• merhaba demek = to say hello

While söylemek also means to say / tell, demek is especially natural when reporting the exact word or phrase someone said.

So günaydın dedim sounds very natural.

Dedim breaks down like this:

• de- = say
• -di = past tense
• -m = I

So:

• de-di-m = I said

Because the verb is demek, its stem here appears as de-.

Because Turkish usually does not need a separate subject pronoun when the verb already shows the person.

• dedim already means I said

So ben is unnecessary unless you want emphasis.

Compare:

• Günaydın dedim = I said good morning
• Ben günaydın dedim = I said good morning

The second version sounds more emphatic or contrastive.

In this sentence, günaydın is the actual expression being said, so it stays as it is.

Turkish often leaves quoted or quoted-like words unchanged after demek:

• Merhaba dedim = I said hello
• Teşekkürler dedim = I said thanks
• Günaydın dedim = I said good morning

So günaydın is not behaving like a normal object that needs extra marking here; it is the content of the speech.

Yes, pretty much.

A very literal breakdown is:

• Öğretmene = to the teacher
• günaydın = good morning
• dedim = I said

Turkish word order is often different from English, and the verb usually comes at the end.

Yes. Turkish word order is flexible, but changing it affects focus or emphasis.

For example:

• Öğretmene günaydın dedim.
  Neutral: I said good morning to the teacher.

• Günaydın öğretmene dedim.
  This can work, but it gives stronger focus to günaydın and may sound less neutral in many contexts.

• Dedim öğretmene günaydın.
  Possible in speech or special contexts, but not the most standard neutral order.

The version you have is a very natural, standard one.

Because -e and -i do different jobs.

• -e / -a = dative, often to
• -i / -ı / -u / -ü = accusative, marking a definite direct object

Here, the teacher is not the direct object. The teacher is the person receiving the greeting, so Turkish uses the dative:

• öğretmene = to the teacher

Not:

• öğretmeni = the teacher as a definite object

Very often, yes, when it means to say something to someone.

Examples:

• Bana bir şey dedi. = He/she said something to me.
• Ona merhaba dedik. = We said hello to him/her.
• Çocuğa adını söyledim. would use söylemek, but with demek you often see the same kind of recipient marking in many contexts.

So with greetings and short expressions, someone + dative + expression + demek is very common.

Not exactly from the Turkish alone.

• öğretmene = to the teacher
• öğretmenime = to my teacher

If the Turkish wanted to clearly say my teacher, it would normally include the possessive ending:

• Öğretmenime günaydın dedim.

Without that ending, öğretmene just means to the teacher. Depending on context, English might still translate it more naturally as to the teacher or sometimes to my teacher, but the Turkish form itself does not explicitly say my.

Yes, very much so.

Turkish commonly forms expressions like:

• günaydın demek = to say good morning
• merhaba demek = to say hello
• hoşça kal demek = to say goodbye
• teşekkür ederim demek = to say thank you

So günaydın dedim is a very normal, idiomatic way to report that greeting.

A rough guide is:

• öğretmene → uh-reht-meh-neh
• günaydın → gew-nai-duhn
• dedim → deh-dim

A few pronunciation notes:

• ö is like the vowel in French peu or German schön
• ü is like French tu or German über
• ğ in öğretmen is usually not pronounced like a hard consonant; it mostly affects the surrounding vowels
• Stress in Turkish is not always as strong as in English, and this whole sentence is usually said fairly smoothly

It would sound unnatural in standard Turkish.

For greetings like günaydın, Turkish strongly prefers demek:

• Öğretmene günaydın dedim. = natural

Using söylemek here is not the usual choice. A learner should treat günaydın demek as the standard pattern.