My sister and her husband have been separated for over 13 months. He is refusing to accept divorce, and she cannot afford to pay for a divorce at this time. They have three kids and she is filing her taxes separately but because she's been supporting the children on her own with no financial support from him, she wants to file Head of Household. Will she legally be able to do this without an official divorce?

As long as they have lived apart for at least the last 6 months of 2013 she can be "considered single," and given that she paid for more than half the upkeep of her home and the children lived with her she may file as Head of Household.

Would she not be audited if he files as married but separated? What would he have to file in order for her not to have any issues? What if he claims one of their children and files head of household too? Sorry for all the questions, she just doesn't want to get into any legal trouble.

How would she prove that he no longer lives there?

There are a number of ways to prove abandonment, such as sworn statement from neighbors or family friends.

An audit is always possible, but it is not likely unless they both claim the same child.