On another lesson using "cabello" I got an answer to your question.
According to this native Spanish speaker, "cabello" is used for the hair on one's scalp and only for humans. "Vello" is hair on the rest of a person's body including one's chin. "Pelo" is used for hair of any kind and on anywhere including animal fur.
Except that Word Reference uses palo in a sentence about hair on ones head: En su juventud, el pelo de mi abuela era de color negro azabache. My grandmother had jet black hair in her youth. https://www.wordreference.com/es/en/translation.asp?spen=pelo
Hair on your head falls under 'hair of any kind and on anywhere,' so as is typical, there are some examples that don't strictly follow the rules. For example, for the color of hair on your head, it's always pelo, never cabello or vello. 'Pelo rubio, pelo negro, pelirrojo, pelo gris,' etc.
"Lavarse" is the infinitive form of the verb. If you wanted to use "lavarse" in this form, you would need to include a properly conjugated form of a verb in front of this. For example:
Ella puede lavarse su cabello. She can wash her hair.
Ella no quiere lavarse su cabello. She doesn't want to wash her hair.
Ella va a lavarse su cabello. She is going to wash her hair.
Hope that helps!
Spanish avoids using possessive pronouns when the verb is reflexive and the owner is perfectly obvious.
Since this is a reflexive statement, we know the subject (ella) owns the hair in question.
Instead of using "su," you use "el cabello".
When you translate it back to English, you need to change "the hair" to "her hair".