Praise be to Allah.
Can you do ghusl without washing hair?
Washing the hair when doing ghusl (full ablution) after having intercourse is obligatory. The evidence for that is the hadith narrated by ‘Aishah who said: When the Messenger of Allah (peace and blessings of Allah be upon him) did ghusl after having intercourse, he would start by washing his hands, then he would pour water with his right hand onto his left hand and wash his private parts, then he would do wudu as for prayer, then he would take the water and put his fingers in his hair till he reached the roots, and when he saw that the water had reached the roots, he would pour water over his head three times. Then he poured water over the rest of his body, then he washed his feet. (Muslim, 474)
Shaykh Ibn Uthaymin said in his commentary on Bulugh al-Maram (p. 399):
“This also indicates that it is obligatory to make the water reach the roots of the hair when doing ghusl after intercourse, and if the hair is thick , one does this by putting one's fingers in one's hair and pouring water over one's head three times after the water has reached the roots of the hair.
Does a woman have to undo her braid when doing ghusl?
This ruling does not differentiate between men and women , but the scholars did discuss undoing the hair if it is braided, in the case of women who do ghusl after intercourse and after finishing their period.
The most correct view is that it is not obligatory to undo braided hair . The evidence for this is the hadith of Umm Salamah, who said: I said, O Messenger of Allah, I am a woman who braids her hair, should I undo it to do ghusl after having intercourse? (According to another report, she said: and after my period?) He said, No. It is enough for you to pour water on your head three times. ( Muslim)
Further evidence is to be seen in the hadith of ‘Ubayd ibn ‘Umayr, who said: ‘Aishah heard that ‘Abd-Allah ibn ‘Umar was commanding women to undo their braids when they did ghusl. She said, “This is very strange on the part of Ibn ‘Umar! He is telling women to undo their braids when they do ghusl. Why doesn’t he tell them to shave their heads?!” The Messenger of Allah (peace and blessings of Allah be upon him) and I used to do ghusl using one vessel, and I never did more than pour water on my head three times. (Muslim, 331)
Shaykh Muhammad ibn Salih al-Uthaymin said in his commentary on Bulugh al-Maram (p. 406):
“This hadith speaks about women doing ghusl: do they have to undo these braids and wash what is underneath them or not? The Messenger (peace and blessings of Allah be upon him) explained that it is not obligatory to undo them every time, because this is too difficult, especially in the case of ghusl after intercourse. There is a difference of scholarly opinion on the matter. Some scholars said that it is obligatory for women to undo their braids when doing ghusl after their periods, but not when doing ghusl after having intercourse. The difference between the two is obviously that ghusl after intercourse is done frequently, and it would be too difficult for a woman to undo her braids in the morning and put them back in the evening or vice versa, but ghusl following one’s period is usually done only once a month, so it is easy for a woman to undo her braids for this purpose.”
And Allah knows best.