Is this sentence with repeated use of "what" correct?

The word which is a relative pronoun used (in many cases) to distinguish one among many: "Which jacket do you prefer?"

The word what is a more generalized term that can refer to anything: "What do you want to do today?"

Therefore, "What experiences, what feelings, what problems does it come from?" is correct because it leaves open for respondents to supply any experiences, feelings, or problems they see fit. To use the word which in your examples would be unnaturally limiting.