И еще:
I am sure that P&G will benefit from hiring me: my various research and analysis skills, as well as my strong leadership and teamwork skills, "укрепившиеся на посту" of the co-leader of student organization make me an excellent candidate.

"укрепившиеся на посту" - и по-русски коряво звучит. Смысл в том, что навыки были и до этого, но тут они развились аж до небывалых высот. =)

"an excellent candidate" - этого достаточно, или лучше an excellent candidate for this position

На счет "мне кажется, невежливо писать про себя "идеальный кандидат" - Если компания западная, очень даже стоит и считается 100% нормально.

Компания очень даже западная.

А первый абзац, который про почему P&G, звучит так:
My name is ///, I am writing to apply for an internship in Consumer and Market Knowledge Department. I would like to work for Procter and Gamble, because is an internship with an organization whose core values are /// would be a great experience for me.

I am positive that I can bring real benefit to P&G; my broad range of research and analytical skills, as well as my strong leadership and teamwork skills, honed during my TIME co-chairing the X, make me an excellent candidate for the position of A

Не надо в начале "Меня зовут".

My two cents:

My name is .... and I'm interested to apply for the internship position with your company in Consumer and Market Knowledge Department. I highly appreciate a chance to receive my internship practice at your company and consider this working experience to be of a great value for me and for my future career.