Is a woman permitted not to wash her hair during ghusl al-janabah if the weather is cold and she fears harm to herself, and if not, is she required to re-perform the prayers she offered with that ghusl, and how should she make them up?

Whoever is required to perform ghusl is not permitted to perform tayammum if water is available and they are able to use it, unless they fear harm or perishing due to extreme cold and are unable to heat the water. In this case, their ruling is like that of someone who cannot find water, and they may perform tayammum and pray. The evidence for this is the action of Amr ibn al-Aas and the Prophet's (peace be upon him) affirmation of it. However, if it is possible to heat the water, then there is no excuse for tayammum. If a woman fears washing her head, then if she can wash and cover it, she must do so. Otherwise, she should perform tayammum for her head and wash the rest of her body.