Basically most of the cases end up so that municipal entities are those providing social housing. And on other side they are also last resort welfare. Generally welfare payments for housing and cost of living is paid directly to accounts of those receiving benefits, but in small number of cases who can not manage their own money it is paid directly from welfare to municipal(more rarely private was these people have really screwed up credit) social housing.
If it's similar to the UK they will get the money for the rent paid to them from the govt, but they have to pay it to the "landlord". In the UK some people can be eligible for direct rent payments to landlords.
Hmm, the article says the landlord is the NGO (unless I've misunderstood). Maybe it's as you say, though.
You pay rent. You get the money to pay the rent from social security benefits.