Huh? I have claimed housing benefit when genuinely unemployed in the past (though not for some years). All they need is proof of address, some form of ID ,a statement of earnings/benefits and details of any dependents etc.
If you operate a cash only business with no paperwork there will always be a temptation to under report earnings.
So yeah, a certain proportion of those who operate cash-only businesses - gardeners, builders, whatever - fraudulently under-report their earnings. A proportion of those will also fraudulently claim benefits, though less than you might think - if you're going to do benefit fraud it makes more sense to do it on a larger scale, you're risking the same penalties and getting more income. But in any case, it's misleading to talk as though this were in any way specific to prostitution.
Others points I believe are correct.