Some cousin will pay hb to a relative. The agent presumably gets a % of the rent as a fee so less than cousin is paying is going to your gm. Cousin gets cheap accommodation , agent gets fee. Yes the utilities should be in cousin's name.
G.mother has dementia which makes it difficult to find out definite details. I believe the contract is with estate agent. just can't get my head round why cousin would leave a larger council house in a better area to move to this house so suddenly. The type of person she is there has to be something in it for her.
Not automatically unless she inherits it 100% in the will but she would be a sitting tenant so whoever owned it would have to follow the relevant legal eviction process. That could be both protracted and potentially acrimonious if it is involves family members. Do you know how long the lease is and the terms of it ?
But gm has dementia - she can't instruct family member to put it up for rent who is receiving the rent? Done any she need to use the income for care home fees? SS will want partial funding surely? Who manages her affairs? Someone has to.....
If your grandmothers savings run out and the rent alone doesn't cover the care home fees I would think the below market rent would raise questions, it would look like they were doing it to stop the house being sold to fund care.