1. You need to be careful that your living arrangements are entirely separate. No joint bank accounts, no joint purchases of food or joint meals, no joint social life, no joint sex life. You would have to be able to prove this to the satisfaction of DWP (& HMRC Tax Credits if she claims them).
2. I'm not sure whether you could charge rent & your ex get Housing Benefit to pay it - the only way of knowing for certain is to enquire of the local authority. But note that if you don't charge, DWP/HMRC may use that as an argument that this is not a commercial arms length arrangement & then be more difficult about the other benefits.