Part 2; please read part 1 first

During my activity there, my practical skills with cases improved and the results have been amazing: over 76% of my patients recovered from depression and now live an a happy life.

1/ I would probably use "work" rather than "activity" there. The work would have involved a range of activities in the plural.

2/ I recommend using "my" to make it clear that you're referring to the  patients that you personally treated. Otherwise, the person who is reading the application might think that you're using "weasel words". That is, you might be saying that 76% recovered, but avoiding saying how much of that was the result of your treatment and taking credit for other people's work.

3/ "An" is only used in in front of vowels or words which start with a vowel sound. "Happy" doesn't; in Italian the "H" is silent in borrowed words (Hotel, Hostess, etc) but it's not in English. H words have a distinct "hhhhuh" sound in front of them.

From 2006 I have been a counselor for the childern of Holy Cross Church in my town. I mainly tutored in school matters and in choosing the future steps in their education.

1/ In English it would be more common to say "Since" rather than "From" here, but in this case From is OK assuming that you're still doing that work. However that does mean that you need to say either "tutor" (present tense) or "have tutored" (past up to now) rather than just "tutored" (past tense). 2/ "Children" has a spelling mistake.

Given my studies, my working experience and my full-day availability, I think I could be a perfect fit for this position.

I would say "would" rather than "could"; the difference is subtle but "would" sounds more confident.

"If you choose me, I WOULD be a perfect fit, I have no doubt" vs "If you choose me, then I could be a perfect fit, maybe, possibly, I'm not sure."

That was well written.

Letter of application

In English it's more conventional to open with "Dear Mr/Ms/Dr/Whatever Surname" rather than with just a statement of the recipient's name.

I would lile like to apply for the position of tutor/assistant for this year's Fair, as advertised in the Psych job newspaper.

I suspect that that was just a typo.

I graduated in Psychology in 2008 at Harvard University and I have always devoted my life to helping people.

I've been an intern at Freud's and Yung's Jung's clinic for 3 years, but actually I am looking for other opportunities and full responsibility positions.

1/ It's pronounced "Yung" (based on the German pronunciation), but spelt "Jung".

2/ "Actually" is often used as a "filler word" in speech, but it rarely adds value in terms of meaning which is why it should be avoided. The only exception is when it is meant to provide a form of contrast with something else. If you take the word out the sentence is still complete, as well as being shorter.

3/ I'm not sure what "full responsibility positions" are; I've never seen "full responsibility" used as an adjective like that. What I think you might mean is:

"... positions where I can take full responsibility for the care and recovery of patients".

Oh for the love of Mike... I'm out of characters again. To be continued.