I'll go through your paragraphs one at a time and show some suggestions. I'll highlight what I'm commenting on in red and provide a number after the text which I will refer to in my suggestions. Some of these are not quite errors, but I might suggest something that sounds better.
(1) You should probably capitalize prof (Prof.) because it is a title.Dear prof (1). mmmmm,
(1) "Theses" is the plural of "thesis." I am guessing you meant "these." A better word would be "this," as in "Thank you very much for providing me with this valuable information."Thank you very much for providing me with theses (1) valuable information. I went(have gone) (2) through your university and department web sites and found many interesting research topics which match my expectation (3). (4) So I'm very much like to (5) commence my career as a graduate student at university of cccccc (6)
(2) Either "went" or "have gone" would be good here. "Went" is shorter, so try "went."
(3) I believe you want a slightly different word than "expectation." An expectation is something you think will happen, so not always what you want to happen. Try "goals," which will better express your desires.
(4) Remember to have a space after this period.
(5) The grammar here is strange to me. Try "As a result, I'm much more likely to commence my career as a graduate student at the university of cccccc."
(6) Remember to put a period here.
(1) "Albeit" is used before a noun. For example, "Albeit the rain, we went outside." If you wish to express this with a clause, use "although." For example, "Although it was raining, we went outside." Remember that a clause is like a little sentence with a subject and verb. Therefore, your sentence should be "That being said, although I'm very much interested about your ..."That being said, albeit (1) I'm very much interested about (2) your faculty and programs (3) I hesitate to apply for (4) the UC for several reasons.
(2) The preposition "in" sounds better here, as in "... I'm very much interested in your faculty and programs..."
(3) You need a comma here because you have a dependent clause, where "I'm very much interested in your faculty and programs" is separate from "I hesitate to apply for the UC for several reasons."
(4) The preposition "to" sounds better here, as in "... I hesitate to apply to the UC for several reasons."
(1) You need a space after the period: "1. Whether my ..."1. (1)whether my qualifications are good enough(Specially GRE score)
(1) This should either be "was a graduate" or "was graduated" or "graduated." I would recommend "graduated." Also remember to put a comma after "....xxxxxx.........."I, ....xxxxxx.......... was a graduated (1) from University of nnnnnn, (2) bbbbbbb with BSc special degree on (3) Molecular biology and Biochemistry obtaining Honors pass and overall GPA of 3.51 (4) (when adjusted to USA system). I have already taken TOEFL iBT test (90) and GRE revised test (Quantitative-153/170, Verbal- 142/170 , Analytical writing -3.5 ). I'm currently working as research coordinator for breast cancer trial at National Cancer Institute , (5)mmmmmm, ssssss, and previously employed as a teaching assistant for undergraduate and MS student at Department of chemistry, Faculty of science, University of bbbbbb, Sri Lanka. (6) I have attached my resume for your perusal, which provides further details about me. (7) As per information provided on UC website, your faculty expecting (8) GRE score of 460 for verbal and 4.0 for analytical writing which clearly above what i have achieved (9).
(2) Be sure to have a space after this comma.
(3) "In" sounds a little better here.
(4) Be sure to have a space before the parenthesis.
(5) Remember to have a space after a comma, and not before. So "... Institute, mmmm, ..."
(6) Be sure to have a space after the period and comma here.
(7) You need a space after this period.
(8) "Expecting" is not a verb when alone. Try "is expecting" or "expects." I would use "expects."
(9) The grammar is strange here. I think you mean "which, as clearly shown above, I have achieved." Also be sure to capitalize the "i."
(1) Be sure to have a space after this period.2. (1) Deadline and fee waiving
(1) Try "I was only able to obtain"I only be able to obtained (1) my official transcript copy this week and I notice (2) that deadline (3) for the international application is 15th (4) which is tomorrow.therefore (5) I would like to know weather (6) your department would accept my application if I send it before 20th (7) and whether I would still able to waive my application fee ? (8)
(2) Use the past-tense. Try "noticed."
(3) You need an article here. Try "the deadline."
(4) Try "the 15th"
(5) You need a space after the period and you need to capitalize "therefore" to "Therefore."
(6) "Weather" refers to state of the sky (like raining, or sunny). You want "whether" here.
(7) Try "the 20th"
(8) Your sentence is not constructed as question, so use a period instead of a question mark.
(1) As a native speaker, this sounds slightly strange. It might sound better to say "I hope you have good news for me."I'm kindly expecting positive reply from you (1).
Thank you very much for your time and consideration (2)
(2) Be sure to have a period.
Be sure to have a comma after "Respectfully"Respectfully (1)
Okay, I hope that helps you! Please do not be discouraged by my corrections. I am impressed by how well you, as someone learning English, can use the language. Also, congratulations on your academic achievements! I hope you receive good news.