Just to clear up any confusion:
My aunt is writing = Η θεία μου γράφει
The aunt is writing to me = Η θεία μού γράφει
When a dative pronoun (to me, to you, etc.) could also be mistaken for a possessive pronoun (my, your, etc.), it is accented. Remember: if a monosyllabic pronoun is accented it is dative.
Source (in Greek): http://ebooks.edu.gr/modules/ebook/show.php/DSDIM-F102/651/4161,19326/ Β7
"The aunt is writing me" is wrong because the translation is "η θεία με γράφει" and not " η θεία μου γράφει". "The aunt is writing" is the correct form and "My aunt is writing" is correct too.
I don't think that's right; English "me" can be used as an indirect object as well.
"She wrote me last week" means that I was the recipient of her message, not that I was the message itself.
You can also say "She wrote to me last week" but you can also say it without the "to".
'Τhe aunt is writing' should be 'Η θεία μού γράφει.' (but the accent is often overlooked).