Few things before I start,
1. You need to tone it down quite a bit. This does not look like a letter of apology at all, and if I, a third person, can read bitterness from this letter so can the very person you're writing to. From the way I see it, this letter will only aggrevate things between you and the recipient of the letter, without some serious editing.
2. Be formal in your writing. By the looks of the letter you're writing to a teacher. Being formal is one way of letting her know that you mean it, that you're serious.
3. English is my second language and I don't guarantee my correction is free of mistakes. Use it at your own discretion.
If I were you I'd re-write the letter from scratch, with a different start and without directly criticizing your teacher.
That said, it's the correction you wants, not my hour-long lecture. Here it goes:
Writer: Sorry to hear that you're gonna stop logging onto this site.
HKB: Sorry to hear that you've decided to stop coming to this site.
(If you want to stick with 'gonna', at least be formal and use 'going to')
Writer: I feel a little guilty cause' we debated so loud and offended each other, I'm sure that could seriously hurt your feelings.
HKB: 1. We had our voices raised and offended each other in a debate. It's mostly my fault and I feel really bad.
2. I feel guilty because I hurt your feeling by raising my voice and offending you in a debate.
(you're 'little' guilty? such notion will only set her off. While you can use either one of the corrections, no.2 is more apologetic.)
Writer: I would like to ask you to stay here and give your precious answers stemed from your huge knowledge of English and to share the way you had studied in foreign country for 18 years.
HKB: I plead you ,in the most sincere manner, to stay here and give your valued answers stemming from your profound knowledge of English and share the way you have studied in foreign countries / a foreign country for 18 years.
(Use past perfect form, had + participe, only when you have more than two different pasts in sequence)
Writer: I did not hate your English-using ability, but your attitude of writing answers to other people's questions, and the style.
HKB: I wasn't criticizing your ability to use English, but the way you write answers to other people's questions.
(Be more specific as to why and how you don't like her way of answering questions. I can't help you much here. If you can't provide the details I suggest you just drop this sentence altogether.)
Writer: I don't understand why in the world you ain't gonna be here any longer. Is that simply because of the stupid insults between you and me?
HKB: (Drop this sentence. If she's offended and hurt by whatever you said or did, it's pretty clear to anyone why. Instead, this is where you need to write your apology.)
Writer: Don't tell me, Maam.
HKB: (Drop this line, too)
Writer: OK, once again, I sincerely apologize to you for all the things that I'd witten to you.
HKB: I sincerely apologize to you for all the things I've written to you.
(Again, this sentence has no past verb - hence 'I've written'.)
Writer: And I also admit that it was so mean and cunning behavior to have revealed the messages that you had posted and the memo that I had sent to you.
HKB: I also admit it was very mean to have revealed the message you posted and the memo I sent to you.
(Cunning doesn't sound too approperiate. Perhaps 'shrewd' suits better here)
HKB's note: I had to go to school by this point and couldn't correct much, but his version seemed fine so I got lazy. Just changed a few words around so from here on it's me + the writer.
In short, I hope to see your (straight-) to-the-point answers here in "naver knowledge web board".
Thank you for reading my mail to the end.
I hope to see your answers again.
p.s.: please open your e-mail and memo so that I can send you a message directly without having to post it in public.