Its a hairnet they put on you when you have a bandage on your head. Long ago they used bandaging around the head like in old war movies (plus in wars they had to work fast with limited resources) but nowadays they use it like this because of the shape of the head (no slipping off this way).
I don't know what to describe him other than an asshole. It's not right to play with a woman's feelings. What would happen if she becomes your girlfriend and the next day at school she sees you holding hands with your childhood friend......... It's better to choose between your childhood friend or your mission as the keeper of the tome. This is modern era, and polygamy isn't a thing anymore.
If you still want to continue the mission then it's better to reject your childhood friend rather than keeping her but hurting her at the sight of you flirting with another woman.
You are totally right @Yuelle. He'd better choose and if he insists on growing stronger, he should let the childhood friend go rather than keeping her in suspense not knowing what their relationship is.