I have read a short story by J. C. Oates. The story tells us about a 15 years old girl, Connie,

In ...year old constructions the years are not plural - it's a very common little mistake 

and her contradictory life. From the one hand, she is a freedom-loving person, from the other hand;

In English the expression is typically "On one hand, ...on the other (hand)" or "On the one hand, on the other..."  (It seems to me that the "the" version seems more common in academic or general American writing)

her life is restricted by her parents. She dreams about escape and an out of duties life,

This is perfectly understandable but very ...russified! Perhaps:

and a life beyond the routine; a life beyond the ordinary; a life out of the ordinary  

but when the opportunity comes, she thinks the matter over. The contradictions pierce her whole life through. The internal conflict of a young girl as we all experienced (it) once. Is it a standard practice for each growing up person?

:D The construction sounds super Russian, but it's only a matter a word order:

for each person, growing up? ; for each of us, growing up?; for each maturing person (perhaps slightly less natural)

I assume «Yes».

To my mind, this story is a kind of The Rorschach test,

a kind of... "uses up" the need for the article.

because every person sees it according to his or her own experience. We find analogy among the lines and events of the story. I suppose, this is the main value of/in it. From my point of view, the short novel is a metaphor, which tells us about a high-time for growing up.

This is a little bit difficult. high-time is usually used in figures of speech only when something is overdue. It has limited use, basically. Perhaps this could be expressed more naturally, depending on your intention:

which tells us about that time when we/one must grow up

which tells us about that moment, when the time has come to grow up

We have to leave everything and set off a new, dangerous journey, which is called “life”.

set off on a new, dangerous journey... or set off anew, on a dangerous journey...

A journey without our parents, without our home and things, which we had got used to living with. Unfortunately, there is no escaping from it. The story is full of symbols, witches are proving it to us.

Actual witches? In which case: full of symbols; the witches prove this to us/ are proof. or on the other hand, The story is full of symbol which prove this to us.

The screen door symbolizes a border between our past and our future. A big, safety house with a fence is our past, our carefree childhood. While two strangers with a jalopy are our inexperienced future and we have to try it. The last passage of the story is the most valuable, to my mind. It gives a full scope to our imagination and our feelings.

The expression to give full scope to is used without the article.

It is a kind of an open end;

Here again, the article is already here earlier, in a kind...

you have a chance to finish it according to your own choice. As I said before, I rather think that land behind Arnold`s back, which Connie has/had not seen before a one more metaphor for the future. The anticipation of fear doesn't(sing.) leave us until the end. Is it that very feeling, which we tend to feel overstepping the line between childhood and the life of a grown up person? Unexplored lands and unknown paths are waiting for us.
The dedication of this story to Bob Dylan is not a coincidence. It is one more proof for my idea about the story. The lines « Leave your stepping stones behind, something calls for you…» and «And it's all over now, Baby Blue» depict once again, that our life is a forward movement (/movement towards something), due to circumstances beyond our control.

In conclusion I want to say that this story is really as controversial as [as... as...] the life of the main character, but despite it/that, I enjoyed reading it and came through all feeling, which it gave to me.  I can almost get the meaning here, could you rephrase this slightly? Perhaps you mean that you felt all of the feelings that were put into the book, or something of that sort.

Sorry if I  missed anything or if the style of the suggestions is a bit awkward. This is in great essay style. Your use of set phrases and figures of speech is excellent.