Is it a correct sentence: What do you think has happened with her?
) it looks like both parts - "you think" and "has happened" require the same word - "what" but here is only one "what".
2) If we give this "what" to "you think" then "has happened" will have no noun.
3).If we will do conversely by giving "what" to "has happened" then "you think" will be left without "what".
4). Or it can be for both of them having only one "what", not two and so on?
Yes it is a correct sentence. The answer is in how we perceive what has been written. From your question and the comments I have read this has been perceived as two phrases, built around "you think" and "has happened". But is this actually the case? The actual meaning of "has happened with" is "occurred relating to"
What do you think has happened with her or What do you think occurred (happened) relating to (with) her. To tidy this up a little.
What do you think occurred relating to her? in this case this is a clearly defined statement not needing a second "what"
For clarity The misconception and confusion regarding the "what's" arises from the focus on "has happened" instead of the more meaningful "has happened with"
Happen means ‘occur’ and most commonly ‘occur by chance’: Cambridge English Dictionary
What will happen if it rains?...............
Will someone tell me what’s happened?
with: preposition (RELATIONSHIP): Cambridge English Dictionary relating to or in the case of a person or thing:
How are things with you?