Yes, your cousin is responsible for business rates for the property. You won't have to make an additional payment to your local council. A commercial property is rated on location and floor space and then business rates are paid on this value. You moving into the same premises wont have a bearing.
Some salons rent out space on a fixed fee, but others also ask for contributions towards Rates, Electricity, Gas, Insurance, Card Machine Charges, Water etc..
As for assuming what you have to pay towards costs. That is entirely up to what you and your cousin have agreed between you.