Is there no way to unambiguously state that a given person has no fixed place of residence, i.e. is "homeless"?-- the only expression I know of is obdachlos sein, which seems to entail sleeping on the street, whereby English homeless does not actually require that one be "sleeping rough" (although this is often the case).
Obdachlos sein can mean that they are sleeping on the street, but it can also be used for people who are just sleeping on someone else couch every night. In my experience though people only think about the first case.
Officially a homeless person is called
Person ohne feste Wohnung (Reference)
Person ohne festen Wohnsitz