Can a father obtain custody of his daughter, who is no more than one year old, from her mother with whom he committed adultery, by filing a lawsuit in the Sharia court, with the presence of witnesses?

If the wife has engaged in an illicit relationship, she cannot be trusted with raising her daughter, and her right to custody is forfeited as long as she remains in that state; because فسق (moral corruption/disobedience to Allah) is one of the impediments to custody. The Sharia judge will rule on the matter based on the evidence presented to him, and the judge's ruling resolves the dispute.