A daughter's tale

Hi all, I would like some advice for my daughter. For the first time at 45 years of age she has had to go on benefit. She is suffering with mental health problems due in part to the stressful role of intensive care nurse, and going through a marriage break up. Some time ago she resigned her position with the NH and did some agency nursing but she stopped working altogether some months ago. At present she is in receipt of ESA contribution based and this is £72 a week. She and her husband are about to put their house on the market, and she will then have a lump sum to hopefully buy her own house. The questions I have are the following. When she gets the money from the sale of her house will the DWP stop paying her the ESA? If she manages to purchase a new home for herself and my grandson (12 years old) will she be entitled to some extra financial help to run the new home? I’m thinking of Council tax and utilities and other costs.