Is it obligatory for a woman to cover her hair when entering the restroom to urinate, for fear of urine splashes that cause baldness, as it was said to be a Hadith?

There is no authentic hadith that commands covering the head when relieving oneself. Some jurists have deemed it recommended, relying on weak hadiths that are not authentic, and on an athar (narration) from Abu Bakr Al-Siddiq, may Allah be pleased with him, that he used to cover his head out of shyness before his Lord. This athar is authentic. In summary, obligation or recommendation is a legal ruling that requires an authentic text for its establishment, and no such text has been reported from the Prophet, peace and blessings be upon him. There is no harm for anyone who does so, following the example of Abu Bakr, may Allah be pleased with him. As for other justifications for covering the head when relieving oneself, some of them are incorrect. The correct justification is shyness before Allah the Almighty.