The se is called a reflexive pronoun which applies to the third person: he, she, it, they, multiple you. Sometime the verbs that require is really are used reflexively as in "Me lavo la cara todos los días." I wash my face everyday. Here, you are washing your own face. If a mother washes the face of her child, the reflexive pronoun would not be used. With levantarse you can think of it as reflexive (I get myself out of bed) but in English we normally just say I get up.
At six means six is the specified time she got up. By six is not focusing on the moment she got up. Instead it means 'no later than'. You could check on her at six and she'd be up. Maybe she got up at three, five, or 5:59. It's irrelevant, because you are talking about her state as of 6, but you're not discussing the moment she got up.