One of the qualifying criteria for Rent Supplement is that a person who has not been renting privately for 6 of the previous 12 months (and been able to afford the rent at the time) must have been assessed as having a housing need by the local authority and be on the housing list - a person who has a property is unlikely to be assessed by the authority as having a housing need.
In a means test, the property is assessed based on its capital value - property value less the outstanding mortgage. Where the house is jointly owned, obviously this value would be halved.