Postacı kadın az önce geldi, mektubu teslim etti.

Turkish has no articles. Definiteness is shown by context and, for direct objects, by case. Here:

• Subject: postacı kadın is understood as definite from context; no article is needed.
• Object: mektubu has the accusative -u, which marks it as definite/specific (“the letter”). If the object were non‑specific, you’d use mektup (no -u) or bir mektup (“a letter”).

Both are possible:

• kadın postacı: more common as “female postie/postwoman,” with the profession as the head noun.
• postacı kadın: “the woman who is a postwoman,” a bit more descriptive or literary; it treats “woman” as the head and “postacı” as a modifier. In casual speech, kadın postacı is generally preferred.

Turkish often links closely related clauses with a comma and no conjunction. You could also say:

• … geldi ve mektubu teslim etti. (perfectly fine)
• … gelip mektubu teslim etti. (using -ip to chain actions with the same subject)

Yes. Time adverbs are flexible:

• Az önce postacı kadın geldi, mektubu teslim etti.
• Postacı kadın az önce geldi, mektubu teslim etti.
• Postacı kadın geldi az önce… (less neutral; can sound emphatic) Positioning affects emphasis/focus rather than grammaticality.

All mean “a short time ago,” with slight nuance:

• az önce: “just now,” very recent.
• biraz önce: “a little earlier,” possibly a bit less immediate.
• demin: colloquial “a moment ago.” All work here.

It’s a light-verb construction: a noun + etmek. Here, teslim etmek = “to deliver/hand over.” You conjugate etmek, not the noun:

• teslim etti (he/she delivered)
• teslim ediyor (is delivering)
• teslim edecek (will deliver)

Sometimes, yes:

• verdi = “gave,” general and informal.
• teslim etti = “delivered/handed over,” more formal/transactional, often implying a duty. In a postal context, teslim etti is the precise choice.

Use the question particle mi/mi̇/ mı/mü as a separate word:

• Postacı kadın az önce geldi mi?
• Mektubu teslim etti mi? For both, ask as two questions, or: Az önce postacı kadın gelip mektubu teslim etti mi?

Subjects aren’t case-marked for definiteness in Turkish. Specific direct objects take the accusative:

• Definite: mektubu teslim etti (“delivered the letter”)
• Indefinite: mektup teslim etti or bir mektup teslim etti (“delivered a letter”)

Yes, use the dative (-a/e):

• Mektubu bana teslim etti. (“[She] delivered the letter to me.”) Common orders: Bana mektubu teslim etti or Mektubu bana teslim etti, depending on what you want to emphasize.

If you didn’t witness it and are reporting/infering, yes:

• Postacı kadın az önce gelmiş, mektubu teslim etmiş. (“Apparently, the postwoman just came and delivered the letter.”) For a narrative anchored in the past, you might see gelmişti/etmişti (“had come/had delivered”).