I will put all the suggested changes in ALL CAPS (so that it will be easier for you to pick out my suggested changes).

Dear Professor Smith,
I hope you are doing very well. I am writing to kindly ask whether you could help ME BY WRITING a reference letter as I HAVE RECENTLY applied for a position in a major international trading company.

The position FOR WHICH I am applying is a Sourcing Manager of Purchasing Dept in a famous US trading company. (DELETED SO) They request THAT candidates provide a reference letter from his/her former UNIVERSITY PROFESSOR. As you are the most prestigious professor in the Faculty of International Trade, (DELETED SO) your reference letter would be GREATLY helpful FOR the employer TO BETTER UNDERSTAND MY QUALIFICATIONS. (DELETED WHAT IS MORE) I hope you (DELETED COULD) still remember that I was one of the top students in class AND always HAD good academic grades.

This is the job that I desire to have because it WILL PROVIDE me the opportunity of personal career development and will enhance my capability of sourcing and interpersonal communication. I WILL also BE offered a chance to get (DELETED SOME) professional training in US.

"So I trust that you letter is very essential to me for getting this job."

(I suggest rewriting this sentence, perhaps something like, "So I assure you that your reference letter is a very essential part for me to get this job")

I would appreciate IT so much if you could reply at your earliest convenience.

Yours Sincerely,

A letter to the a former professor ( I would appreciate if someone you could check for me. Thanks)

Dear Professor Smith,

It had been a long while since I have graduated.  I Hope you are doing very that all is well. I am writing to kindly ask whether you could help write me a reference letter as recently I have applied for a position in a major international trading company. It would be great if you can give me a letter of reference.

The position I am applying for is the position a of sourcing manager of in the purchasing dept in of a famous [don't use such adjectives.  It tells the reader that you are showing off, especially to your superiors] US trading firm. company. So they request candidates to provide One requirement is to obtain a reference letter from his/her former professor in university. As you are the most prestigious professor in the Faculty of International Trade, so your reference letter would be great helpful to the employer who can understand about my qualification. [Too direct and too much ass kissing! You won't get a letter from me. :) ] Of all the faculty, I cannot think of a better professor and teacher who had a better mould and guide for me.  For this I thank you. What is more, hope you could still remember that I was one of the top students in class with always good academic grades. [This is definitely a no-no! Either he remembers you or he don't! Just leave this out.  If you REALLY want to use something like this, use it as something like hearted like, "In case you forget, I am the girl in pigtails with a lisping tooth" or "... the girl who always sit in the front row etc" something that tells you apart from others]

This is the job that I desire to have because it provides me the opportunity of personal career development and will enhance my capability of sourcing and interpersonal communication. I would be also offered a chance to get some professional training in US. So I trust that you letter is very essential to me for getting this job. [画蛇添足! and also in a very demanding tone!  So omit all this!]

I would appreciate so much if you could reply at your earliest convenience. Your kindness is most appreciated.  I look forward to your reference letter.  Once again, I thank you for not forgetting this girl in pigtails with a lisping tooth. [repeat whatever outstanding feature you had.  This gives an air of closeness, warmth and light heartedness.  Also, this will set your letter apart from others and your professor will remember you better.  Also keep your letter short!]

Yours Sincerely former student [親切些],

A letter to the former professor ( I would appreciate if someone could check(it) for me )

Dear Professor Smith,

(Long time no contact)I hope you are doing very well. I am writing to kindly ask whether you could help write me me write a reference letter as recently I have applied I am applying for a position in a major international trading company.

The position I am applying is a Sourcing Manager of Purchasing Dept in a famous US trading company. So they request candidates( it's better to use "so the candiates were asked )

 to provide a reference letter from his/her former professor in university. As you are the most prestigious professor in the Faculty of International Trade, so your reference letter would be great helpful to the employer who can understand about my qualification. What is more, hope you could still remember that (still I believe you could remember) I was one of the top students in class with always good academic grades.

This is the job that I desire to have because it provides me a good oppurtunity for my future career development the opportunity of personal career development and will enhance my capability of sourcing and interpersonal communication+skills. I would be also (also I will be offered a good chance)offered a chance to get some professional training in US + once employed. So I trust that you letter is very essential to me for me getting this job.

I would appreciate so much if you could reply me at your earliest convenience.

Yours Sincerely,

++++ 

suggestions: Next time you email a foreign professor you should be more straightforward, and to be confident. In this way you can make them trust you well. You can use a polite tone but more importantly you should express your intention more directly.