Is the young woman considered sinful, and has she wronged her husband? Is her prayer permissible and acceptable despite not speaking to him? Is it permissible for her to go out without informing him, knowing that she cannot bear to live with him after the marriage contract?

If the marriage contract is valid and meets the conditions, then there is no blame upon you for leaving the house without your husband's permission as long as he is not providing for you. As for abandoning him and not speaking to him, it is not permissible to continue this unless there is a legitimate شرعي justification. If you dislike him and fear neglecting his rights, then you may seek a divorce in exchange for compensation that you pay to him.